\section{Introduction}

The Ising model plays a central role in equilibrium statistical mechanics, being a standard example of an order-disorder phase transition in dimensions
two and above. Besides purely mathematical interest, its studies have found successful applications to ferromagnetism, lattice gases,
chemical adsorbates, ecology and image processing.

The phase transition in the Ising model in two dimensions has been a subject of extensive study, both in mathematics and physics literature.
Onsager \cite{onsager} exactly computed the free energy and the critical exponents of the model. Further, a gradual understanding of the Ising model at criticality led to the conjecture by Belavin, Polyakov and Zamolodchikov that its scaling limit (as well as scaling limits of other critical models) should be conformally invariant and described by Conformal Field Theory \cite{belavin-polyakov-zamolodchikov-i,belavin-polyakov-zamolodchikov-ii}.

In the mathematical language, this conjecture may be formulated as follows. Consider a continuous planar domain $\Omega$, and let $\Omega_\delta$ be discrete approximations to this domain. Then, as the mesh size $\delta$ tends to zero, some quantities (expectations, probabilities etc.) in the model, under a proper normalization, have conformally invariant or covariant limits. Moreover, Conformal Field Theory predicts \emph{exact formulae} for these limits.

The mathematical proofs of such predictions were out of reach until recently. In 2006, Smirnov \cite{smirnov-i} has rigorously established conformal invariance of the \emph{fermionic observables} in the Ising model. Later, this result has been proven to be universal \cite{chelkak-smirnov-ii}, and led to the proof of convergence of the interfaces in the Ising model to Schramm's SLE${}_3$ curves \cite{CDHKS}. The full-plane \emph{energy} correlation functions (that is, the correlations of $n$ pairs of neighboring spins) were computed on periodic isoradial graphs by Boutillier and de Tili\`ere \cite{boutillier-de-tiliere-i,boutillier-de-tiliere-ii}, and the scaling limit for bounded domains has been rigorously treated in \cite{hongler-smirnov-ii,hongler-i}, confirming the CFT predictions for the energy field.

The corresponding question about the \emph{spin} correlations remained open, and the existing results mainly concern the case of the full plane. The diagonal and horizontal spin-spin correlations in the plane were computed by Wu \cite{mccoy-wu-i}. Recently, Pinson \cite{pinson} has proven the rotational invariance of the full-plane two-point correlations. A number of remarkable results were obtained for the massive limits, see \cite{wu-mccoy-tracy-barouch,sato-miwa-jimbo-iv,palmer-tracy} and references therein. Palmer \cite{palmer} justified the CFT predictions at criticality by taking the zero-mass limit. In the presence of boundary, however, even the simplest problem of the asymptotics of magnetization of the Ising model in the half-plane remained open since it was posed by McCoy and Wu \cite{mccoy-wu-ii}.

In this paper, we rigorously prove existence and conformal covariance of scaling limits of all multi-point spin correlation functions in any simply-connected planar domain with $+$ boundary conditions. For example, our result for magnetization, when combined with the asymptotics of the two-point correlations in the plane \cite{mccoy-wu-i}, reads as follows. Let $\E_{\Omega_\delta}^{+}[\sigma_a]$ denote the magnetization (expectation of spin) at the cite $a$ with $+$ boundary conditions in a discrete domains $\Omega_\delta$. Then
\[
\delta^{-\frac{1}{8}}\E_{\Omega_\delta}^{+}[\sigma_a] \to \mathcal{C}_1\cdot\text{rad}^{-\frac{1}{8}}(a,\Omega),
\]
as $\Omega_\delta$ approximates $\Omega$ and the mesh size $\delta$ tends to zero. Here $\mathcal{C}_1$ is an explicit lattice-dependent constant, and $\text{rad}(a,\Omega)$ denotes the conformal radius of $\Omega$ as seen from $a$, or, in other words, $\text{rad}(a,\Omega)=|\varphi'(0)|$, where $\varphi$ is a conformal map from the disc $\{z\in\C:|z|<1\}$ to $\Omega$ mapping the origin to $a$.

We establish similar results for the multi-point correlations $\E_{\Omega_\delta}^{+}[\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}]$, and, in the case of two-point function, for free boundary conditions (see Theorems \ref{thm:2pts} and \ref{thm:1-3pts}). Our previous results \cite{chelkak-izyurov} immediately allow one to treat  alternating $+$/$-$ boundary conditions as well (see Remark \ref{rem: alt_bc}). The method we use also applies to mixed correlations, involving spin, energy, disorder and boundary change operators, and extends to multiply-connected domains, which will be worked out in a subsequent paper.

The explicit formulae for the the scaling limits of $\E_{\Omega_\delta}^{+}[\sigma_{a_0}\sigma_{a_1}\dots,\sigma_{a_k}]$ were derived by the Conformal Field Theory methods in \cite{belavin-polyakov-zamolodchikov-i,cardy-i,burkhard-guim}. In our approach the answers come in an explicit, but different form, being defined via solutions to a special interpolation problem, so one should additionally check that they coincide with the known CFT predictions. In this paper, we do this check for $k=1,2,3$ and for the case when all the points are on the same vertical line in the half-plane (see further discussion in Sections~\ref{sub:12pts-computations} and the Appendix), which gives all the properties of the continuous correlations we need in the proof.

Our main tool is discrete holomorphic fermionic observables. We use the spinor version of those, introduced in \cite{chelkak-izyurov}. Fermionic observables per se were essentially invented back in 70's \cite{kadanoff-ceva}, and reminiscents of the spinor versions can be found in the works of Kyoto's school \cite{sato-miwa-jimbo-i,sato-miwa-jimbo-ii,sato-miwa-jimbo-iii,sato-miwa-jimbo-iv}. However, rigorous proofs of convergence require the delicate analysis of discrete boundary value problems for the observables, developed more recently \cite{smirnov-i,smirnov-ii,chelkak-smirnov-ii}.

Simultaneously and independently of our work, Dub\'edat derived the 2n-point spin correlation functions in the full plane at criticality,
combining exact bosonization techniques \cite{dubedat-ii} and results on monomer correlations in the dimer model \cite{dubedat-i}. Analogous results have been announced for polygonal domains.

\subsection{Main results} The Ising model on a graph $\mathcal{G}$ is a random assignment of
$\pm1$ spins to the vertices $\mathcal{V}$ of $\mathcal{G}$. The
probability of a spin configuration $\sigma\in\left\{ \pm1\right\} ^{\mathcal{V}}$
is proportional to
\[
e^{-\beta\mathbf{H}\left(\sigma\right)},
\]
where $\beta>0$ is the inverse temperature and $\mathbf{H}\left(\sigma\right):=\sum_{x\sim y}\sigma_{x}\sigma_{y}$
is the energy of the configuration. We will consider the Ising model in discrete planar domains $\Omega_\delta$ which are subsets of the refining \emph{square grids} rotated by 45\textdegree{} of diagonal mesh sizes $2\delta$ (thus, the distance between adjacent spins is $\sqrt{2}\delta$, see Figure~\ref{Fig:discrete-domain}). In this paper we are interested in the setup when $\Omega_\delta$ approximate some simply connected domain $\Omega$ as $\delta\to 0$.

If we consider the Ising model on the \emph{faces} of $\Omega_\delta$ with $+$ boundary conditions (the spins on the boundary of $\Omega$ are set to $+$), there is a natural mapping (known as the low-temperature expansion, see \cite[Chapter 1]{palmer}) between the spin configurations and the collections of closed
contours (interfaces) on $\Omega_{\delta}$: trace an edge between any two faces with different spins. Under this mapping, the probability of a collection of interfaces $\omega\subset\Omega_{\delta}$ becomes proportional to
\[
\alpha^{\#\mathrm{edges}\left(\omega\right)},
\]
where $\alpha=\exp\left(-2\beta\right)$. The same procedure applied to free boundary conditions (no restrictions are set for boundary spins) produces collections of contours which also may start and end on the boundary of $\Omega$.

From now on we only consider the model at its \emph{critical point}, which for the square grid corresponds to the parameter value $\beta_{c}=\frac{1}{2}\ln\left(\sqrt{2}+1\right)$, or equivalently to $\alpha_{c}=\sqrt{2}-1$, when the phase transition occurs. As a normalizing factor, we will use the spin correlation in the unique infinite-volume limit of the critical planar Ising model: set
\[
\varrho\left(\delta\right):=\mathbb{E}_{\mathbb{C}_{\delta}}[\sigma_{0_\delta}\sigma_{1_\delta}],
\]
where $\mathbb{C}_{\delta}$ denotes the square grid $\delta (1\!+\!i)\mathbb{Z}^2$, while $0_\delta$ and $1_\delta$ stand for proper approximations of the points $0,1\in\mathbb{C}$. We first state the convergence theorem for the two-points functions, both with $+$ and free boundary conditions:

\begin{thm}\label{thm:2pts}
Let $\Omega$ be a bounded simply connected domain and $\Omega_\delta$ be discretizations of $\Omega$ by the square grids of diagonal mesh size $2\delta$. Then, for any $\epsilon>0$, we have
\begin{eqnarray*}
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}{\varrho(\delta)} \; \to \; \left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{+}\quad\text{and}\quad
\frac{\mathbb{E}_{\Omega_{\delta}}^{\mathrm{free}}\left[\sigma_{a}\sigma_{b}\right]}{\varrho(\delta)} \; \to \; \left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{\mathrm{free}},
\end{eqnarray*}
as $\delta\to 0$, uniformly over all $a,b\in\Omega$ at distance at least $\epsilon$ from $\partial\Omega$ and from each other, where $\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{+}$ and $\left\langle \sigma_{a}\sigma_{b}\right\rangle_{\Omega}^{\mathrm{free}}$ are explicit functions given by (\ref{eq: conf_conv_2}) and (\ref{2ptFcts}).
\end{thm}
\begin{rem}
(i) Here and below we follow the traditional notation developed by physicists and denote the limits of spin correlations by the symbols $\langle \sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^\mathfrak{b}$ which should be understood as \emph{functions} of the points $a_0,\Ak\in\Omega$ which may also depend on $\Omega$ and boundary conditions $\mathfrak{b}$ as parameters.

\smallskip

\noindent (ii) The functions $\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{+}$ and $\left\langle \sigma_{a}\sigma_{b}\right\rangle_{\Omega}^{\mathrm{free}}$ are conformal covariants of degree $\frac{1}{8}$
with respect to $a$ and $b$: for any conformal map $\varphi:\Omega\to\Omega'$, one has
\begin{equation}
\left\langle \sigma_{a}\sigma_{b}\right\rangle_{\Omega}= \langle\sigma_{\varphi(a)}\sigma_{\varphi(b)}\rangle_{\Omega'}\cdot|\varphi'\left(a\right)|^{\frac{1}{8}}||\varphi'\left(b\right)|^{\frac{1}{8}}.
\label{eq: conf_conv_2}
\end{equation}

\noindent (iii) A celebrated result of T. T. Wu \cite{mccoy-wu-i} gives the rate of decay of $\varrho(\delta)$:
\[
\varrho\left(\delta\right)\sim \mathcal{C}_2\cdot\delta^{\frac{1}{4}},\quad \delta\to 0,
\]
where the lattice dependent constant is given by $\mathcal{C}_2=2^{\frac{1}{3}}e^{-3\zeta'\left(-1\right)}$, and
$\zeta'$ denotes the derivative of Riemann's zeta function. However, we do not need this result in our proofs.
\end{rem}
We have a similar result for multi-point correlations with $+$ boundary conditions:

\begin{thm}\label{thm:1-3pts}
Let $\Omega$ be a bounded simply connected domain and $\Omega_\delta$ be discretizations of $\Omega$ by the square grids of diagonal mesh size $2\delta$. Then, for any $\epsilon>0$ and any $k=0,1,\dots$, we have
\begin{eqnarray*}
\varrho(\delta)^{-\frac{k+1}{2}}\cdot \mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\right] & \to & \left\langle \sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\right\rangle _{\Omega}^{+}
\end{eqnarray*}
as $\delta\to 0$, uniformly over all $a_0,a_1,\dots,a_k\in\Omega$ at distance at least $\epsilon$ from $\partial\Omega$ and from each other, and the functions $\langle\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ have the following covariance under conformal mappings $\varphi:\Omega\to\Omega'$:
\begin{equation}
\label{conf-cov}
\langle\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}=\langle \sigma_{\varphi(a_0)}\sigma_{\varphi(a_1)}\dots\sigma_{\varphi(a_k)}\rangle _{\Omega'}^{+} \cdot {\textstyle\prod_{j=0}^k}|\varphi'\left(a_j\right)|^{\frac{1}{8}}.
\end{equation}
\end{thm}

The convergence of $(k+1)$-point correlation with free boundary conditions (which by symmetry is only non-zero for odd $k$) can be obtained by our techniques as well, but for simplicity we do not include this in the present paper.

Due to conformal covariance (\ref{eq: conf_conv_2}), (\ref{conf-cov}), it is sufficient to give the explicit expressions for the correlation functions in the upper half-plane $\bH$. For magnetization and two-point functions, they read
\begin{eqnarray}
\label{1ptFct} \left\langle \sigma_{a}\right\rangle_{\mathbb{H}}^{+} & = & \frac{2^\frac{1}{4}}{(2\Im\mathfrak{m}a)^{\frac{1}{8}}},\\ \label{2ptFcts}
\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\mathbb{H}}^{+} & = & \frac{\sqrt{u_{ab}+u_{ab}^{-1}}}{(2\Im\mathfrak{m}a)^{\frac{1}{8}}(2\Im\mathfrak{m}b)^{\frac{1}{8}}},\qquad  \left\langle \sigma_{a}\sigma_{b}\right\rangle _{\mathbb{H}}^{\mathrm{free}} ~=~ \frac{\sqrt{u_{ab}^{-1}-u_{ab}}}{(2\Im\mathfrak{m}a)^{\frac{1}{8}}(2\Im\mathfrak{m}b)^{\frac{1}{8}}}, \quad\text{where}~~u_{ab}:=\left|\frac{b-a}{b-\overline{a}}\right|^\frac{1}{2}.
\end{eqnarray}
A different way to write these formulae involves conformal radius $\text{rad}(a,\Omega)$ and hyperbolic distance $\mathrm{d}_{\Omega}^{\mathcal{H}}$:
\begin{eqnarray*}
\left\langle \sigma_{a}\right\rangle _{\Omega}^{+} & = & 2^\frac{1}{4}\mathrm{rad}\left(a,\Omega\right)^{-\frac{1}{8}},\\
\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{+} & = & \frac{\left\langle \sigma_{a}\right\rangle _{\Omega}^{+}\left\langle \sigma_{b}\right\rangle _{\Omega}^{+}}{\left(1-\exp\left(-2\mathrm{d}_{\Omega}^{\mathcal{H}}\left(a,b\right)\right)\right)^{\frac{1}{4}}}, \qquad
\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{\mathrm{free}}  ~=~  \frac{\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{+}}{\exp\left(\frac{1}{2}\mathrm{d}_{\Omega}^{\mathcal{H}}\left(a,b\right)\right)}.
\end{eqnarray*}
For the discussion of the explicit form of $\langle\sigma_{a_0}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$, $k\geq 2$, we refer the reader to the Appendix.

\begin{rem}[\emph{alternating boundary conditions}]
Using results of \cite{chelkak-izyurov}, one can immediately generalize Theorem~\ref{thm:1-3pts} to the alternating $+/-$ boundary conditions. Namely, let $\mathfrak{b}=\left\{ b_{1},\ldots,b_{2n}\right\}$ be a collection of points on $\partial\Omega$ appearing in counterclockwise
order and let $\mathfrak{b}_{\delta}\subset\partial\Omega_{\delta}$ denote a proper discretization of the family $\mathfrak{b}$. Let $\mathbb{E}_{\Omega_{\delta}}^{\mathfrak{b}}$ denote an expectation for the Ising model (as above, defined on the faces of $\Omega_\delta$) with the $+$ boundary conditions on the counterclockwise arcs $\left[b_{2j-1},b_{2j}\right]$ and $-$ boundary condition on the complementary arcs $\left[b_{2j},b_{2j+1}\right]$, $j=1,\dots,n$ (we set $b_{2n+1}:=b_1$). It was proven in \cite{chelkak-izyurov} that all the ratios $\mathbb{E}_{\Omega_{\delta}}^{\mathfrak{b}}[\sigma_{a_0}{\dots}\sigma_{a_k}]/\mathbb{E}_{\Omega_{\delta}}^{+}[\sigma_{a_0}{\dots}a_k]$ have explicit conformally invariant scaling limits $P_\Omega^\mathfrak{b}(a_0,{\dots},a_k)$. Thus, Theorem~\ref{thm:1-3pts} implies that the expectations $\varrho(\delta)^{-\frac{k+1}{2}}\mathbb{E}_{\Omega_{\delta}}^{\mathfrak{b}}[\sigma_{a_0}{\dots}\sigma_{a_k}]$ themselves have conformally covariant scaling limits
\[
\langle \sigma_{a_0}{\dots}\sigma_{a_k}\rangle_\Omega^\mathfrak{b}:=P_\Omega^\mathfrak{b}(a_0,{\dots},a_k)\cdot \langle \sigma_{a_0}{\dots}\sigma_{a_k}\rangle_\Omega^+.
\]
\label{rem: alt_bc}
\end{rem}
\subsection{Key steps in the proof}\label{sub:key-theorems}
In this section we list the key results that allow us to prove Theorems~\ref{thm:2pts} and \ref{thm:1-3pts}. We start our analysis with the \emph{discrete logarithmic derivatives} of the spin-spin correlations with $+$ boundary conditions. The following Theorem, which identifies the ratios of those correlation for neighboring faces is a cornerstone for the whole paper (keep in mind that our square lattice is rotated by 45\textdegree):
\begin{thm}
\label{thm:log-derivatives}
Let $\Omega$ be a bounded simply connected domain and $\Omega_\delta$ be discretizations of $\Omega$ by the square grids $\delta(1\!+\!i)\mathbb{Z}^2$. Then, for any $\epsilon>0$ and any $k=0,1,\dots$, we have
\begin{eqnarray}
\frac{1}{2\delta}\left(\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a+2\delta}\sigma_{a_1}\dots\sigma_{a_k}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]}-1\right) & \to & \Re\mathfrak{e}\mathcal{A}_\Omega(a;a_1,\dots,a_k),
\label{eq:log-der-x}\\
\frac{1}{2\delta}\left(\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a+2i\delta}\sigma_{a_1}\dots\sigma_{a_k}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]}-1\right) & \to & -\Im\mathfrak{m}\mathcal{A}_\Omega(a;a_1,\dots,a_k)
\label{eq:log-der-y}
\end{eqnarray}
as $\delta\to 0$, uniformly over all faces $a,a_1,\dots a_k\in\Omega_{\delta}$ at distance at least $\epsilon$ from $\partial\Omega$ and from each other. The function $\mathcal{A}_\Omega(a;a_1,\dots,a_k)$ is defined explicitly via solution to some special interpolation problem (see further details in Section~\ref{sub:cont-spinors}) and has the following covariance property under conformal mappings $\varphi:\Omega\to\Omega'$:
\begin{equation}
\label{A-covariance}
\mathcal{A}_\Omega(a;a_1,\dots,a_k)= \mathcal{A}_{\Omega'}(\varphi(a);\varphi(a_1),\dots,\varphi(a_k))\cdot\varphi'(a)+
\frac{1}{8}\frac{\varphi''(a)}{\varphi'(a)}.
\end{equation}
\end{thm}
\begin{proof}
The proof is based on the convergence results for the discrete spinor observables. A rather delicate analysis is needed since we are interested in the values of observables near their singular points. See further details in Sections~\ref{sec:spinors-and-correlation}~and~\ref{sec:analysis-of-spinors}.
\end{proof}

Now one can reconstruct the quantities $\log\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_\Omega^+$ by integrating their derivatives $\mathcal{A}_\Omega(a;a_1,\dots,a_k)$ inside $\Omega$. In particular, the constants in the integration can be chosen so that the explicit formulae (\ref{1ptFct}) and (\ref{2ptFcts}) given above hold true. In Section~\ref{sub:decorrelations} we also discuss how to choose those multiplicative constants coherently for all $\Omega$ and $k$ in order to give a consistent definition of the continuous correlation functions. Note that the conformal covariance degree $\frac{1}{8}$ in (\ref{conf-cov}) is a direct consequence of the covariance rule (\ref{A-covariance}) for $\mathcal{A}_\Omega(a;a_1,\dots,a_k)$ (see Remark~\ref{rem: covariance-ratios}).

Theorem~\ref{thm:log-derivatives} allows us to prove the following weaker form of the convergence result for the spin correlations:
\begin{cor}
\label{cor:conv-up-to-normalization}
Under conditions of Theorem~\ref{thm:log-derivatives}, for any $k\ge 0$ there exist some normalizing factors $\varrho_{k+1}(\delta,\Omega_\delta)$ that might depend on $\Omega$ and the number of points $a,a_1,\dots,a_k$ but not on their positions such that
\[
\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right] ~\sim~ \varrho_{k+1}(\delta,\Omega_\delta)\cdot \left\langle \sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right\rangle _{\Omega}^{+}
\]
as $\delta\to 0$, uniformly over all faces $a,a_1,\dots a_k\in\Omega_{\delta}$ at distance at least $\epsilon$ from $\partial\Omega$ and from each other.
\end{cor}
\begin{proof}
See Section~\ref{sub:decorrelations}.
\end{proof}

We now focus on the special case $k=1$, i.e. on the two-point functions. The next theorem is a crucial tool which allows us to compare the normalizing factors $\varrho_2(\delta,\Omega_\delta)$ with the full-plane case. We denote by $\mathbb{E}_{\Omega_{\delta}^\bullet}^{\mathrm{free}}$ the expectation for the critical Ising model defined on the \emph{vertices} of $\Omega_\delta$ (with free boundary conditions).
\begin{thm}
\label{thm:ratio-free-plus}
Let $\Omega$ be a bounded simply connected domain and $\Omega_\delta$ be discretizations of $\Omega$ by the square grids $\delta(1\!+\!i)\mathbb{Z}^2$.
Then, for any $\epsilon>0$, we have
\[
\frac{\mathbb{E}_{\Omega_{\delta}^\bullet}^{\mathrm{free}}\left[\sigma_{a+\delta}\sigma_{b+\delta}\right]}{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]} ~\underset{\delta\to0}{\longrightarrow}~
\mathcal{B}_{\Omega}(a;b),
\]
uniformly over all faces $a,b$ at distance at least $\epsilon$ from $\partial\Omega$ and from each other, where $\mathcal{B}_{\Omega}(a;b)$ is a conformal invariant of $(\Omega,a;b)$ which can be explicitly written via two-point functions (\ref{2ptFcts}) as $\mathcal{B}_{\Omega}(a;b)={\left\langle \sigma_{a}\sigma_{b}\right\rangle_{\Omega}^{\mathrm{free}}}/{\left\langle \sigma_{a}\sigma_{b}\right\rangle_{\Omega}^{+}}$.
\end{thm}
\begin{proof}
The proof is based on the convergence results for the discrete spinor observables and the Kramers-Wannier duality,
see further details in Sections~\ref{sub:discrete-spinors-to-ratios-corr},~\ref{sub:convergence-of-spinors} and Section~\ref{sec:analysis-of-spinors}.
\end{proof}

The FKG inequality gives $\mathbb{E}_{\Omega_{\delta}^\bullet}^{\mathrm{free}}\left[\sigma_{a+\delta}\sigma_{b+\delta}\right]\le \mathbb{E}_{\mathbb{C}_{\delta}}\left[\sigma_{a}\sigma_{b}\right]\le \mathbb{E}_{\Omega_{\delta}}^{\mathrm{+}}\left[\sigma_{a}\sigma_{b}\right]$ and it is easy to check that $\mathcal{B}_{\Omega}(a;b)\to 1$ as $b$ approaches to $a$. Since the normalizing factors $\varrho_2(\delta,\Omega_\delta)$ do not depend on the positions of $a,b\in\Omega$, we conclude that $\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]\sim \mathbb{E}_{\mathbb{C}_{\delta}}\left[\sigma_{a}\sigma_{b}\right]$ in the double limit when $\delta\to 0$ and (later on) $b\to a$, thus relating $\varrho_2(\delta,\Omega_\delta)$ with the full-plane normalization and establishing Theorem~\ref{thm:2pts} (see further details in Section~\ref{sub:ratios-to-thm-2pts}).

Having the universal two-point function normalization $\varrho_2(\delta,\Omega_\delta)\sim\varrho(\delta)$ as $\delta\to 0$, we derive asymptotics of all other $\varrho_{k+1}(\delta,\Omega_\delta)$ using the following observation: as the point $a$ approaches $\partial\Omega$, the continuous correlation functions are defined so that one has
\begin{equation}
\label{eq: dec-identity-intro}
\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_\Omega^+\sim \langle\sigma_a\rangle_\Omega^+ \langle\sigma_{a_1}\dots\sigma_{a_k}\rangle_\Omega^+
\end{equation}
and the same decorrelation result $\E_{\Omega_\delta}^{+}[\sigma_a\sigma_{a_1}\dots\sigma_{a_k}]\sim\E_{\Omega_\delta}^{+}[\sigma_a]\E_{\Omega_\delta}^{+}[\sigma_{a_1}\dots\sigma_{a_k}]$ holds true in the double limit $\delta\to 0$ and $a\to\partial\Omega$ (see details in Section~\ref{sub:proof-thm-k-pts}). This implies the recurrent formula $\varrho_{k+1}(\delta,\Omega_\delta)\sim \varrho_1(\delta,\Omega_\delta)\varrho_k(\delta,\Omega_\delta)$ for $k=1,2,\dots$ and, further, $\varrho_{k+1}(\delta,\Omega_\delta)\sim\varrho(\delta)^{(k+1)/2}$ for all $k$, thus establishing Theorem~\ref{thm:1-3pts}.


\subsection{Organization of the paper}
The rest of this paper is devoted to the proof of Theorems~\ref{thm:2pts} and~\ref{thm:1-3pts}. Section~\ref{sec:spinors-and-correlation} outlines the proof and contains all the main ideas. Details, especially those involving hard discrete complex analysis techniques, are mostly postponed to Section~\ref{sec:analysis-of-spinors}. The reader not interested in these details may restrict himself to Section~\ref{sec:spinors-and-correlation} only.

We fix the notation in Section~\ref{sub:notation-and-definitions}. The main tool of this paper, the discrete holomorphic spinor observables, is introduced and discussed in Sections~\ref{sub:discrete-spinors} and~\ref{sub:s-holomorphicity}. In Section~\ref{sub:discrete-spinors-to-ratios-corr}, we prove that the ratios of spin correlations that appear in Theorems~\ref{thm:log-derivatives} and~\ref{thm:ratio-free-plus} can be expressed in terms of these observables. In Sections~\ref{sub:cont-spinors} and~\ref{sub:convergence-of-spinors}, we discuss the continuous counterparts of the discrete observables and state the convergence Theorems \ref{thm:obs-away},~\ref{thm:localization-near-a} and~\ref{thm:localization-near-b}, that easily imply Theorems \ref{thm:log-derivatives} and~\ref{thm:ratio-free-plus}. We derive the formulae~(\ref{1ptFct}) and~(\ref{2ptFcts}) in Section~\ref{sub:12pts-computations}, and prove Corollary~\ref{cor:conv-up-to-normalization} in Section~\ref{sub:decorrelations}. We complete the proofs of Theorems~\ref{thm:2pts} and~\ref{thm:1-3pts} in Sections~\ref{sub:ratios-to-thm-2pts} and~\ref{sub:proof-thm-k-pts}, respectively.




Section~\ref{sec:analysis-of-spinors} is devoted to the proof of Theorems \ref{thm:obs-away},~\ref{thm:localization-near-a} and~\ref{thm:localization-near-b}. We discuss the discrete properties of our observables and their full-plane analogue in Sections~\ref{sub:s-holomorphicity-proof}--\ref{sub: riemann-problem}, and finish the proof of the main convergence theorems in Sections~\ref{sub:convergence-away-sing} and~\ref{sub:convergence-near-sing}. In the Appendix, we explain how to compute explicitly the continuous spinor observables in the half-plane. We also check that the correlation functions coincide with the CFT predictions at least when all points are on the same vertical line in the upper half-plane, and prove decorrelation identity~(\ref{eq: dec-identity-intro}).

\subsection*{Acknowledgements}
All three authors wish to thank Stanislav Smirnov for introducing them to the Ising model and discrete complex analysis, for sharing his ideas and for constant support. We also would like to thank St\'ephane Benoist, David Cimasoni, Julien Dub\'edat, Hugo Duminil-Copin, Pierluigi Falco, Christophe Garban, Barry McCoy, Nikolai Makarov, Charles Newman, John Palmer, Yuval Peres, Mikhail Sodin, Tom Spencer and Yvan Velenik for many helpful discussions and comments.

The work of Dmitry Chelkak and Konstantin Izyurov was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St.-Petersburg State University) under RF Government grant 11.G34.31.0026. Dmitry Chelkak was partly supported by P.Deligne's 2004 Balzan prize in Mathematics.
Cl\'ement Hongler was supported by the National Science Foundation under grant DMS-1106588 and by the Minerva Foundation.
Konstantin Izyurov was partially supported by ERC AG CONFRA and the Swiss National Science Foundation.



\section{\label{sec:spinors-and-correlation}Holomorphic Spinors and Correlation Functions}

\subsection{\label{sub:notation-and-definitions}Notations} We start by fixing the notation which is used throughout the paper.

\begin{figure}

\centering{\includegraphics[height=0.33\textheight]{fig-discrete-domain.eps}}

\caption{\label{Fig:discrete-domain} An example of a discrete domain $\Omega_\delta$ and notation for the sets of vertices ($\mathcal{V}_{\Omega_\delta}^\bullet$), faces ($\mathcal{V}_{\Omega_\delta}^\circ$), edge midpoints ($\mathcal{V}_{\Omega_\delta}^\bullet$) and (four types) of corners ($\mathcal{V}_{\Omega_\delta}^\mathrm{c}=\mathcal{V}_{\Omega_\delta}^\mathrm{1}\cup\mathcal{V}_{\Omega_\delta}^\mathrm{i}\cup \mathcal{V}_{\Omega_\delta}^\mathrm{\lambda}\cup\mathcal{V}_{\Omega_\delta}^\mathrm{\overline{\lambda}}$). The mesh size $\delta$ is a half-diagonal of
a square face, thus the distance between adjacent spins is $\sqrt{2}\delta$. The inner vertices, faces, edges and corners are colored black, while the boundary ones are colored gray.}
\end{figure}


\subsubsection{Graph notation}
Recall that we work on the square grid rotated by 45\textdegree{}
\[
\mathbb{C}_{\delta}:= \left\{\delta\left(1+i\right)\left(m+in\right):m,n\in\mathbb{Z}\right\}.
\]
The mesh size $\delta$ is hence the size of a half-diagonal of a square face. We often will identify the vertices of $\mathbb{C}_{\delta}$ with the corresponding complex numbers, the faces of $\mathbb{C}_{\delta}$ with their centers, the edges of $\mathbb{C}_{\delta}$ with their midpoints, etc.

We call \emph{discrete domain of mesh size $\delta$} a union of grid faces (see also Figure~\ref{Fig:discrete-domain} for notation given below). We say that $\Omega_{\delta}$ is \emph{simply connected} if the corresponding polygonal domain is simply connected.

\begin{itemize}
\item We denote by ${\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\circ}}$ the set of all \emph{faces} belonging to $\Omega_{\delta}$ (which are identified with their centers), by ${\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\bullet}}$ the set of all \emph{vertices} incident to these faces, and by ${\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}}$ the set of all \emph{edges} incident to $\mathcal{V}_{\Omega_{\delta}}^{\circ}$, which are identified with their midpoints (or the vertices of a medial lattice).
\end{itemize}

In order to simplify the presentation, we also assume that \emph{$\Omega_\delta$ has no fiords of a single face width}, i.e., all the edges joining vertices from $\mathcal{V}_{\Omega_{\delta}}^{\bullet}$ belong to $\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}$. This technical assumption can be easily relaxed, if necessary.

\begin{itemize}
\item We denote by ${\partial\mathcal{V}_{\Omega_{\delta}}^{\circ}}$, ${\partial\mathcal{V}_{\Omega_{\delta}}^{\bullet}}$ and ${\partial\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}}$ the sets of \emph{boundary faces, vertices and edges}, i.e. those faces, vertices and edges which are incident but do not belong to $\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\circ}$, $\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\bullet}$ and $\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}$, respectively (see Figure~\ref{Fig:discrete-domain}).
\item We set $\mathcal{V}_{\Omega_{\delta}}^{\circ}:=\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\circ}\cup\partial\mathcal{V}_{\Omega_{\delta}}^{\circ}$ etc.
\end{itemize}

Below we also need to work with four \emph{corners} of a given square face separately (see the crucial Definition~\ref{def:discrete-spinor} below). For a given vertex $v\in\mathbb{C}_\delta$, we identify the nearby corners with the points $v\pm\frac{1}{2}\delta$ and $v\pm\frac{1}{2}\delta i$ on the complex plane.
\begin{itemize}
\item We denote by ${\mathcal{V}_{\Omega_{\delta}}^{\mathrm{c}}}$ the set of all corners incident to the vertices from $\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\circ}$. We also set ${\mathcal{V}_{\Omega_{\delta}}^{\mathrm{cm}}}:=\mathcal{V}_{\Omega_{\delta}}^{\mathrm{c}}\cup\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}$.
\end{itemize}

\subsubsection{Double covers}\label{sub:double-covers}
In this paper we often deal with holomorphic functions (both discrete and continuous) which are defined on a double cover of a planar domain $\Omega$ and changes the sign between sheets (i.e., have the multiplicative monodromy $-1$ around branching points). We call such functions \emph{holomorphic spinors}. The following notation will be used below:
\begin{itemize}
\item For a planar domain $\Omega$ and $a\in\Omega$, we denote by ${\left[\Omega,a\right]}$
the double cover of $\Omega\setminus\left\{ a\right\}$  branching around $a$. All such double covers are naturally
viewed as subdomains of $\left[\mathbb{C},a\right]$. We often identify points on a double cover with their base points (so each base point is identified with the two points above it).
\item For several marked points $a_1,\dots,a_k\in\Omega$, we denote by $\left[\Omega,a;a_1,\dots,a_k\right]$
   
    the double cover of $\Omega\setminus\{a,a_1,\dots,a_k\}$ which branches around each of the points $a,a_1,\dots,a_k$.
\item We will often compare spinors defined on $\left[\Omega,a;a_1,\dots,a_k\right]$ with those defined on $[\C,a]$ (e.g., with 1/$\sqrt{z-a}$ or $\sqrt{z-a}$) near the common branching point $a$. We write such equations meaning that they are valid in a small $\epsilon$-neighborhood of $a$, with the natural correspondence of sheets.
\end{itemize}
We will also consider double covers of discrete domains. In this case the following slightly modified notation will be convenient:
\begin{itemize}
\item For a discrete domain $\Omega_{\delta}$ and a \emph{face} $a\in\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\circ}$, we set $\left[\Omega_{\delta},a^{\rightarrow}\right]:=\left[\Omega_{\delta},a\right]\setminus\left\{a+\frac{\delta}{2}\right\}$, excluding both points over the corner $a+\frac{\delta}{2}$ from the natural double cover branching at $a$. Similarly, we set $\left[\Omega_{\delta},a^{\rightarrow};\{a_j\}\right]:=\left[\Omega_{\delta},a;\{a_j\}\right]\setminus\left\{a+\frac{\delta}{2}\right\}$, if several additional faces $a_1,\dots,a_k\in\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\circ}$ are marked.
\end{itemize}

\subsubsection{Contours}\label{sub:contours}

In order to define the main tool of the paper -- discrete holomorphic spinors, we need some additional notation related to the contour representation of the Ising model:
\begin{itemize}
\item We denote by $\mathcal{C}_{\Omega_{\delta}}$ the family of all collections of closed
contours on $\Omega_{\delta}$, i.e. the family of subsets of edges $\omega\subset\mathcal{V}_{\Omega_{\delta}}^\mathrm{m}$
such that every vertex $v\in\mathcal{V}_{\Omega_{\delta}}^{\bullet}$ belongs to an even number of edges in $\omega$.
\end{itemize}
Recall that $\mathcal{C}_{\Omega_{\delta}}$ is in a natural $1$-to-$1$ correspondence with the spin configurations of the Ising model on the faces of $\Omega_\delta$ with $+$ boundary conditions (all the spins corresponding to boundary faces $\partial\mathcal{V}_{\Omega_{\delta}}^{\circ}$ are set to $+$): trace an edge between any two faces with different spins. Below we also introduce the families of contour collections which, besides a number of closed loops, contain a single path running from one fixed corner $x$ to another corner or an edge midpoint $y$:
\begin{itemize}
\item For $x,y\in\mathcal{V}_{\Omega_{\delta}}^{\mathrm{cm}}$, let $\pi_{x,y}=xv_{1}\ldots v_{n}y$
be some walk with $v_{1},\ldots,v_{n}\in\mathcal{V}_{\Omega_{\delta}}^{\bullet}$.
We set $\mathcal{C}_{\Omega_{\delta}}\left(x,y\right):=\{\omega\oplus\pi_{x,y},\omega\in\mathcal{C}_{\Omega_{\delta}}\}$,
where $\oplus$ denotes the XOR, or symmetric difference. It is easy to see that $\mathcal{C}_{\Omega_{\delta}}\left(x,y\right)$ does not depend on the particular choice of $\pi_{x,y}$. Note that, for any  $\gamma\in\mathcal{C}_{\Omega_{\delta}}\left(x,y\right)$, there exists a (non-unique) decomposition of $\gamma$ into a collection of disjoint, simple loops and a path  $\mathrm{p}\left(\gamma\right)\subset\gamma$ running from $x$ to $y$. By a decomposition we mean that each edge in $\gamma\in\mathcal{C}_{\Omega_{\delta}}\left(x,y\right)$ belongs to exactly one loop (or to $p(\gamma)$) and is visited only once, and that there are no transversal intersections or self-intersections (see Figure~\ref{Fig:spinor}).
\end{itemize}

\subsection{\label{sub:discrete-spinors}Construction of discrete spinor observables}

Now we are ready to introduce discrete spinor observables. The following definition generalizes the construction given in \cite{chelkak-izyurov} to the case when a ``source'' point is inside $\Omega_\delta$.

\begin{defn}\label{def:discrete-spinor} Let $\Omega_{\delta}$ be a discrete domain and $a,a_1,\dots,a_k\in\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\circ}$
be inner faces. For a corner $z\in\mathcal{V}_{\left[\Omega_{\delta},a^{\rightarrow};a_1,\dots,a_k\right]}^{\mathrm{c}}$ (below we also extend this definition to edge midpoints, see Remark~\ref{rem:spinor-def}(iii)), we define
\begin{equation}
\label{spinor-corners-def} F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\right]}\left(z\right) ~:=~ \frac{1}{\mathcal{Z}_{\Omega_{\delta}}^+\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{ a_k}\right]}\sum_{\gamma\in\mathcal{C}_{\Omega_{\delta}}\left(a+\frac{\delta}{2},z\right)}\alpha_{\mathrm{c}}^{\#\mathrm{edges}\left(\gamma\right)}
\cdot\phi_{a;a_1,\dots,a_k}\left(\gamma,z\right),
\end{equation}
where
\begin{itemize}
\item $\#\mathrm{edges}\left(\gamma\right)$ is the number of full
edges contained in $\gamma$ and $\alpha_{\mathrm{c}}=\sqrt{2}-1$
as above;
\item the complex phase $\phi_{a;a_1,\dots,a_k}(\gamma,z)$ is defined by (see also Figure~\ref{Fig:spinor})
\[
\phi_{a;a_1,\dots,a_k}\left(\gamma,z\right):= e^{-\frac{i}{2}\mathrm{wind}\left(\mathrm{p}\left(\gamma\right)\right)} \cdot \left(-1\right)^{\#\mathrm{loops}_{a;a_1,\dots,a_k}\left(\gamma\setminus\mathrm{p}\left(\gamma\right)\right)}\cdot \mathrm{sheet}_{a;a_1,\dots,a_k}\left(\mathrm{p}\left(\gamma\right),z\right),
\]
where, for a decomposition of $\gamma$ mentioned in Section~\ref{sub:contours},
\begin{itemize}
\item $\mathrm{wind}\left(\mathrm{p}\left(\gamma\right)\right)$ is
the total winding of the path $\mathrm{p}\left(\gamma\right)$ when going from
$a+\frac{\delta}{2}$ to $z$, that is, the number of left turns minus the number of right turns, multiplied by $\frac{\pi}{2}$.
\item $\#\mathrm{loops}_{a;a_1,\dots,a_k}\left(\gamma\setminus\mathrm{p}\left(\gamma\right)\right)$
is the number of loops in $\gamma\setminus\mathrm{p}\left(\gamma\right)$
that contain an odd number of marked points $a,a_1,\dots,a_k$ inside (equivalently, that change the sheet of the double-cover $[\Omega_\delta,a;a_1,\dots,a_k]$),
\item the last factor $\mathrm{sheet}_{a;a_1,\dots,a_k}\left(\mathrm{p}\left(\gamma\right),z\right)$
is equal to $+1$ if $z$ is on the same sheet of $\left[\Omega_{\delta},a;a_1,\dots,a_k\right]$ as the lift of the end of $\mathrm{p}\left(\gamma\right)$,
and to $-1$ otherwise (more precisely, we fix one of the two points lying over the ``source'' $a+\frac{\delta}{2}$ once forever and identify all other $z\in[\Omega_\delta,a;a_1,\dots,a_k]$ with paths running from \emph{this} $a+\frac{\delta}{2}$ to $z$ modulo homotopy and an appropriate index $2$ subgroup of the fundamental group);
\end{itemize}
\item the normalizing factor $\mathcal{Z}_{\Omega_{\delta}}^+\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]$
is defined by
\begin{eqnarray*}
\mathcal{Z}_{\Omega_{\delta}}^+\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right] & := & \sum_{\omega\in\mathcal{C}_{\Omega_{\delta}}}\alpha_{\mathrm{c}}^{\#\mathrm{edges}\left(\omega\right)} \left(-1\right)^{\#\mathrm{loops}_{a;a_1,\dots,a_k}\left(\omega\right)}.
\end{eqnarray*}
\end{itemize}
\end{defn}

\begin{figure}

\centering{\includegraphics[height=0.33\textheight]{fig-spinor.eps}}

\caption{\label{Fig:spinor} A contour collection $\gamma\in\mathcal{C}_{\Omega_\delta}(a+\frac{\delta}{2},z)$ decomposed into non-intersecting loops (dashed) and a path $\mathrm{p}(\gamma)$. Running from $a+\frac{\delta}{2}$ to the projection of $z$, this path makes a $3\pi$ turn counterclockwise, thus $e^{-\frac{i}{2}\wind(\mathrm{p}(\gamma))}=-i$. There is a single loop in $\gamma$ surrounding $a$, hence $(-1)^{\mathrm{loops}_a(\gamma\setminus \mathrm{p}(\gamma))}=-1$. Being lifted to the double cover $[\Omega_\delta,a]$, this path ends on the other sheet, thus $\mathrm{sheet}_a(\mathrm{p}(\gamma),z)=-1$, and $\phi_a(\gamma,z)=-i$.}
\end{figure}

\begin{rem}\label{rem:spinor-def}
(i) For any collection of closed contours $\omega\in\mathcal{C}_{\Omega_{\delta}}$, the sign $(-1)^{\#\mathrm{loops}_{a;a_1,\dots,a_k}\left(\omega\right)}$ coincides with the product of spins $\sigma_a\sigma_{a_1}\dots\sigma_{a_k}$ in the corresponding Ising model configuration with $+$ boundary values. Since $\alpha_{\mathrm{c}}^{\#\mathrm{edges}\left(\omega\right)}$ is just the Ising weight of $\omega$, one concludes that
\[
\mathcal{Z}_{\Omega_{\delta}}^+\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]= \mathbb{E}_{\Omega_{\delta}}^+\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]\cdot \mathcal{Z}_{\Omega_{\delta}}^+,
\]
where $\mathcal{Z}_{\Omega_{\delta}}^+=\sum_{\omega\in\mathcal{C}_{\Omega_{\delta}}}\alpha_{\mathrm{c}}^{\#\mathrm{edges}}$ is the partition function of the model. In particular, $\mathcal{Z}_{\Omega_{\delta}}^+\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]>0$.

\smallbreak

\noindent (ii) It is easy to check that the complex phase $\phi_{a;a_1,\dots,a_k}\left(\gamma,z\right)$ is independent of the choice of the decomposition of $\gamma$ into a path $p(\gamma)$ and a collection of loops, e.g., see discussion in~\cite{chelkak-izyurov}. Note that there are four types of corners: lying to the right of a nearby vertex $v$, below $v$, to the left of $v$, and upper $v$. For each of these groups, the total turning of the path $\mathrm{p}(\gamma)$ is defined uniquely modulo $2\pi$. Therefore, the discrete spinors introduced above always have purely real values at the first group corners, are collinear to $\lambda:=e^{\frac{\pi}{4}i}$ for the second group, etc. This motivates the following notation:
\begin{itemize}
\item we partition the set $\mathcal{V}_{\mathbb{C}_{\delta}}^{\mathrm{c}}$ of all corners into four subsets $\mathcal{V}_{\Omega_{\delta}}^{1}$, $\mathcal{V}_{\Omega_{\delta}}^{\lambda}$, $\mathcal{V}_{\Omega_{\delta}}^{i}$ and $\mathcal{V}_{\Omega_{\delta}}^{\overline{\lambda}}$ depending on the position of a nearby vertex $v\in\mathcal{V}_{\mathbb{C}_\delta}^{\bullet}$ (to the left, upper, to the right, below) with respect to the corner.
\end{itemize}
(iii) \emph{We extend Definition~\ref{def:discrete-spinor} to edge midpoints $z\in\mathcal{V}_{\left[\Omega_{\delta},a^{\rightarrow};a_1,\dots,a_k\right]}^{\mathrm{m}}$ by adding the factor $(\cos\frac{\pi}{8})^{-1}$ to the formula (\ref{spinor-corners-def})}, with $\#\mathrm{edges}\left(\gamma\right)$ being the number of full edges contained in $\gamma$ and the complex phase $\phi_{a;a_1,\dots,a_k}(\gamma,z)$ being defined as above. Note that in this case each edge midpoint $z$ can be reached by a path $\mathrm{p}(\gamma)$ from two opposite sides. Hence, the sum in (\ref{spinor-corners-def}) contains both real and imaginary terms, thus producing a complex number as the spinor value $F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\right]}\left(z\right)$.
\end{rem}

\subsection{S-holomorphicity and boundary conditions}
 \label{sub:s-holomorphicity}
 A version of discrete holomorphicity, the notion of \emph{s-holomorphicity} was introduced in \cite{smirnov-i} together with the nonbranching version of discrete holomorphic observables, as a tool to study the critical Ising model on the square lattice. The properties of such functions were further investigated in \cite{chelkak-smirnov-ii} for a more general class of graphs. On the square grid, s-holomorphic functions may be thought of as (more classical) discrete holomorphic functions whose real part is defined on $\mathcal{V}^{1}_{\Omega_\delta}$ and imaginary part on $\mathcal{V}^{i}_{\Omega_\delta}$, extended in a particular way to $\mathcal{V}^{\lambda}_{\Omega_\delta}$,~$\mathcal{V}^{\overline{\lambda}}_{\Omega_\delta}$, and further to $\mathcal{V}^{\mathrm{m}}_{\Omega_\delta}$ (see more details in Section~\ref{sub:s-holomorphicity-proof}). Our definitions resemble those in \cite{smirnov-i}.

\begin{defn}
\label{def: shol}
With each corner $x\in\mathcal{V}_{\mathbb{C}_{\delta}}^{\tau}$ (with $\tau\in\left\{ 1,i,\lambda,\overline{\lambda}\right\} $),
we associate the line $\ell\left(x\right):=\tau\mathbb{R}$ in the complex plane, and denote by $\mathsf{P}_{\ell\left(x\right)}$ the projection onto that line, defined by
\[
\mathsf{P}_{\ell\left(x\right)}\left[w\right]:=\tfrac{1}{2}\left(w+\tau^{2}\overline{w}\right),\quad w\in\C.
\]
We say that a function $F:\mathcal{V}_{\Omega_{\delta}}^{\mathrm{cm}}\to\mathbb{C}$
is \emph{s-holomorphic in $\Omega_\delta$} if for every $x\in\mathcal{V}_{\Omega_{\delta}}^{\mathrm{c}}$
and $z\in\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}$ that are adjacent, one has
\[
F\left(x\right)=\mathsf{P}_{\ell\left(x\right)}\left[F\left(z\right)\right].
\]
For functions defined on double covers, we introduce the notion of s-holomorphicity exactly in the same manner.
\end{defn}

The following proposition contains the crucial properties of the observables $F_{[\Omega_\delta,a;a_1,\dots,a_k]}$ that will allow us to analyze their scaling limits. For $z\in \partial\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}$, let $\nu_{\mathrm{out}}(z)$ denote the \emph{``outer normal to the boundary at $z$''}: the edge whose midpoint is $z$, oriented towards the exterior of the domain and viewed as a complex number.
\begin{prop}
\label{prop: s-hol+bc}
 The function $F_{[\Omega_\delta,a;a_1,\dots,a_k]}$ is s-holomorphic and has (multiplicative) monodromy $-1$ around each of the marked points $a,a_1,\dots,a_k$, thus being a discrete s-holomorphic spinor on $\mathcal{V}_{\left[\Omega_{\delta},a^{\rightarrow};a_1,\dots,a_k\right]}^{\mathrm{cm}}$. Also,
\begin{equation}
\label{eq: discrete_bc}
\Im\mathfrak{m}\,\left[F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\,\right]}\left(z\right){\sqrt{\nu_{\mathrm{out}}(z)}}\right]=0~~\text{for~all~~} z\in\partial\mathcal{V}_{\left[\Omega_{\delta},a^{\rightarrow};a_1,\dots,a_k\right]}^{\mathrm{m}}.
\end{equation}
\end{prop}
\begin{proof}
We give a proof (which is mainly based on the standard XOR bijection, cf.~\cite{chelkak-smirnov-ii}) in Section~\ref{sub:s-holomorphicity-proof}.
\end{proof}

\begin{rem}
\label{rem:intF^2}
The boundary conditions (\ref{eq: discrete_bc}) are a priori not robust enough to pass to the scaling limit: even if the limiting domain $\Omega$ has a smooth boundary, the discrete normal $\nu_{\mathrm{out}}\left(z\right)$ can possibly admit only the values $\sqrt{2}e^{\pm\frac{\pi i}{4}}$ and $\sqrt{2}e^{\pm\frac{3\pi i}{4}}$, and so does not (pointwise) converge to its continuous counterpart. These conditions become much nicer, if one finds a way ``to integrate'' the square of $F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\right]}$: the real part of this integral satisfies Dirichlet boundary conditions on $\partial\Omega$ due to (\ref{eq: discrete_bc}). This approach is not as straightforward, since the square of a discrete holomorphic function is, in general, not discrete holomorphic, and so does not have a well-defined discrete primitive. However, it was noted in~\cite{smirnov-i} that one can naturally define the real part of the integral, using the \emph{s-holomorphicity} of discrete observables, which is a stronger property than the usual discrete holomorphicity. Moreover, a technique developed in \cite{chelkak-smirnov-ii} allows one to treat this real part essentially as if it were a harmonic function, see further details in Section~\ref{sub: riemann-problem}.
\end{rem}


\subsection{\label{sub:discrete-spinors-to-ratios-corr}From discrete spinors
to ratios of correlations}
The following lemma expresses the ratios of spin correlations (or their discrete logarithmic derivatives) from Theorem~\ref{thm:log-derivatives} in terms of the spinor observables introduced in Section~\ref{sub:discrete-spinors}. Moreover, it also provides the crucial ingredient for the proof of Theorem~\ref{thm:ratio-free-plus}.

\begin{lem}
\label{lem:ratios-disc-magnet-spinor} For any $k=0,1,\dots$, we have
\begin{eqnarray}
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a+2\delta}\sigma_{a_1}\dots\sigma_{a_k}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]} & = & F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\right]}\left(a+\tfrac{3\delta}{2}\right),\label{eq:log-diff-spinor-hor}
\end{eqnarray}
where we take the corner $a+\frac{3\delta}{2}$ on the same sheet as the ``source'' point $a+\frac{\delta}{2}$.
Moreover, for $k=1$, we also have
\begin{eqnarray}
\frac{\mathbb{E}_{\Omega_{\delta}^\bullet}^{\mathrm{free}}\left[\sigma_{a+\delta}\sigma_{b+\delta}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]} & = & \pm i F_{\left[\Omega_{\delta},a;b\right]}\left(b+\tfrac{\delta}{2}\right),\label{eq:spinor-duality}
\end{eqnarray}
where $\mathbb{E}_{\Omega_{\delta}^\bullet}^{\mathrm{free}}$ denotes the expectation for the critical Ising model defined on the vertices of $\Omega_\delta$ (with free boundary conditions outside the set $\mathcal{V}_{\Omega_{\delta}}^{\bullet}$) and the sign $\pm$ depends on the sheet where the corner $b+\frac{\delta}{2}$ is taken.
\end{lem}
\begin{proof}
Recall that
\begin{eqnarray*}
F_{\left[\Omega_{\delta},a;a_1\dots, a_k\right]}\left(a+\tfrac{3\delta}{2}\right) & = & \frac{\sum_{\gamma\in\mathcal{C}_{\Omega_{\delta}}\left(a+\frac{\delta}{2},a+\frac{3\delta}{2}\right)} \alpha_{\mathrm{c}}^{\#\mathrm{edges}\left(\gamma\right)}\phi_{a;a_1,\dots,a_k}\left(\gamma,a+\frac{3\delta}{2}\right)} {\mathcal{Z}_{\Omega_{\delta}}^+\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]},
\end{eqnarray*}
while, taking into account Remark~\ref{rem:spinor-def}(i),
\begin{eqnarray*}
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a+2\delta}\sigma_{a_1}\dots\sigma_{a_k}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]} & = & \frac{\sum_{\omega\in\mathcal{C}_{\Omega_{\delta}}}\alpha_{\mathrm{c}}^{\#\mathrm{edges} \left(\omega\right)}\left(-1\right)^{\#\mathrm{loops}_{a+2\delta;a_1,\dots,a_k}(\omega)}} {\mathcal{Z}_{\Omega_{\delta}}^+\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]}.
\end{eqnarray*}
There exists an obvious bijection between the sets $\mathcal{C}_{\Omega_{\delta}}\left(a+\frac{\delta}{2},a+\frac{3\delta}{2}\right)$ and $\mathcal{C}_{\Omega_{\delta}}$: removing the two corner-edges $(a,a+\frac{\delta}{2})$ and $(a+\delta,a+\frac{3\delta}{2})$ from a given contour $\gamma$, we obtain a contour $\omega(\gamma)\in\mathcal{C}_{\Omega_{\delta}}$ and vice versa. So, it suffices to show that
\[
\phi_{a;a_1\dots,a_k}\left(\gamma,a+\tfrac{3\delta}{2}\right)=\left(-1\right)^{\#\mathrm{loops}_{a+2\delta;a_1\dots,a_k}\left(\omega(\gamma)\right)}.
\]
Let us pick any loop in $\gamma$ and remove it. The left-hand side (respectively, the right-hand side) has changed the sign if and only if there was an odd number of points $a,a_1,\dots,a_k$ (respectively, $a+2\delta,a_1,\dots,a_k$) inside the loop. However, no loop in $\gamma$ separates $a$ form $a+2\delta$ (such a loop would intersect $p(\gamma)$), so the two sides can only change sign simultaneously. Thus it is sufficient to consider the case when $\gamma$ is just a single non-self-intersecting path $\mathrm{p}(\gamma)$ running from $a+\frac{\delta}{2}$ to $a+\frac{3\delta}{2}$, which is treated by the following observations:  $\text{sheet}_{a;a_1\dots,a_k}(p(\gamma),a+\frac{3\delta}{2})=-1$ if and only if there is an odd number of points $a,a_1,\dots,a_k$ inside the loop $\omega(\gamma)$, and $\mathrm{wind}(p(\gamma))=2\pi\mod 4\pi$ if an only if $\omega(\gamma)$ separates $a$ from $a+2\delta$.

For (\ref{eq:spinor-duality}), the Kramers-Wannier duality (e.g., see \cite[Chapter 1]{palmer}) implies
\[
\frac{\mathbb{E}_{\Omega_{\delta}^\bullet}^{\mathrm{free}}\left[\sigma_{a+\delta}\sigma_{b+\delta}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}= \frac{\sum_{\omega\in\mathcal{C}_{\Omega_{\delta}}(a+{\frac{\delta}{2}},b+\frac{\delta}{2})}\alpha_{\mathrm{c}}^{\#\mathrm{edges}\left(\omega\right)}} {\mathcal{Z}_{\Omega_\delta}^+[\sigma_a\sigma_b]},
\]
hence it is sufficient to prove that the (purely imaginary) number $\phi_{a;b}\left(\gamma,b+\frac{\delta}{2}\right)$ does not depend on $\gamma$. We have $\#\mathrm{loops}_{a;b}(\gamma\setminus\mathrm{p}(\gamma))=0$, since any loop in $\gamma$ either surrounds both $a,b$ or none of them (otherwise it would intersect the path $\mathrm{p(\gamma)}$ joining $a+\frac{\delta}{2}$ and $b+\frac{\delta}{2}$). Further, let $\pi_{ba}^\circ=bv_1\dots v_ma$ be some simple walk with $v_1,\dots,v_m\in \mathcal{V}^{\circ}_{\Omega_\delta}$. Then, $p(\gamma)\cup(b+\frac{\delta}{2},b)\cup\pi_{ab}^\circ\cup(a,a+\frac{\delta}{2})$ is a loop, which we denote by $l(\gamma)$. Let $n(\gamma)$ be the number of self-intersections of $l(\gamma)$ (i.e., the number of intersections of $\mathrm{p}(\gamma)$ with $\pi_{ba}^\circ$). Since $\exp[-\frac{i}{2}\mathrm{wind}(l(\gamma))]=(-1)^{n(\gamma)+1}$ (this is true for any closed loop), we see that
\[
\exp[-\tfrac{i}{2}\mathrm{wind}(\mathrm{p(\gamma)})]=(-1)^{n(\gamma)+1} \cdot \exp[\tfrac{i}{2} \mathrm{wind}((b+\tfrac{\delta}{2},b)\cup\pi_{ba}^\circ\cup(a,a+\tfrac{\delta}{2}))],
\]
where the second factor does not depend on $\gamma$. But we may also view $\pi_{ba}^\circ$ as a cut defining a sheet of the double cover $[\Omega,a;b]$, meaning that $\text{sheet}_{a;b}(p(\gamma),b+\frac{\delta}{2})=(-1)^{n(\gamma)}$, which proves the desired result.
\end{proof}

\begin{rem}
\label{rem:ratios-disc-magnet-spinor} Similarly to (\ref{eq:log-diff-spinor-hor}), one can check that
\begin{eqnarray}
\label{eq:nearby-ratios-disc-magnet-spinor}
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a+\left(1\pm i\right)\delta}\sigma_{a_1}\dots\sigma_{a_k}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]} & = & e^{\pm\frac{\pi i}{4}}\: F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\right]}\left(a+\left(1\!\pm\!\tfrac{i}{2}\right)\delta\right). \label{eq:log-diff-spinor-diag}
\end{eqnarray}
The proof boils down to the identity
\[
e^{\pm\frac{\pi i}{4}}\phi_{a;a_1,\dots,a_k}\left(\gamma,a+\left(1\!\pm\!\tfrac{i}{2}\right)\delta\right) =
\left(-1\right)^{\#\mathrm{loops}_{a+\delta(1\pm i);a_1,\dots,a_k}\left(\omega(\gamma)\right)}
\]
for the natural bijection $\gamma\mapsto\omega(\gamma)$ removing the two corner-edges $(a,a+\frac{\delta}{2})$ and $(a+\delta,a+(1\!\pm\!\frac{i}{2})\delta)$ from a given contour $\gamma\in\mathcal{C}_{\Omega_{\delta}}\left(a+\frac{\delta}{2},a+(1\pm\frac{i}{2})\delta\right)$, which we leave to the reader.
\end{rem}

\subsection{\label{sub:cont-spinors}Continuous spinors} In this Section, we introduce the continuous counterparts of the discrete spinor observables defined in Section~\ref{sub:discrete-spinors}: the continuous holomorphic spinors $f_{[\Omega,a;a_1,\dots,a_k]}$. We define them as
solutions to the conformally covariant Riemann boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}), which is a continuous analogue of the corresponding discrete boundary value problem (see Remark~\ref{rem:cont-spinor-motivation} below).

\begin{defn}
Let $\Omega$ be a bounded simply-connected domain with smooth boundary, and $a,a_1,\dots,a_k\in\Omega$.
We define $f_{\left[\Omega,a;a_1,\dots,a_k\right]}$ to be the (unique) holomorphic spinor on $[\Omega,a;a_1,\dots,a_k]$, branching around each of $a,a_1,\dots,a_k$ and satisfying the following conditions:
\begin{eqnarray}
\Im\mathfrak{m}\,\left[f_{[\Omega,a;a_1,\dots,a_k]}(z)\sqrt{\nu_\mathrm{out}(z)}\,\right]&=&0,~~z\in\partial \Omega;\label{def:spinor-bdry}\\
 \lim\limits_{z\rightarrow a_j}\sqrt{z-a_j}\cdot f_{[\Omega,a;a_1,\dots,a_k]}(z)&\in&i\R; \label{def:spinor-1}\\
\lim\limits_{z\rightarrow a}\sqrt{z-a}\cdot f_{[\Omega,a;a_1,\dots,a_k]}(z)&=&1. \label{def:spinor-2}
\end{eqnarray}
where $\nu_\mathrm{out}(z)$ denotes the outer normal to the boundary of $\Omega$ at $z$.
\label{def:spinor-bounded}
\end{defn}

\begin{rem}
\label{rem: obs_uniqueness}
\label{rem: spinor_rough}
(i) Note that a solution to the boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}) is unique, if exists. Indeed, if $f_1,f_2$ denote two different solutions, then the spinor $f_1-f_2$ satisfies two first conditions, while $\lim_{z\rightarrow a}\sqrt{z-a}\cdot (f_1(z)-f_2(z))=0$. Applying the Cauchy residue theorem to the \emph{single-valued} function $(f_1(z)-f_2(z))^2$, one arrives at
\begin{equation}
\label{(f1-f2)^2-int}
0\le i^{-1}\int_{\partial\Omega}(f_1(z)-f_2(z))^2dz=2\pi \sum_{j=1}^k \lim\limits_{z\rightarrow a_j}(z-a_j)(f_1(z)-f_2(z))^2\le 0,
\end{equation}
where the first inequality easily follows from (\ref{def:spinor-bdry}) and the second from (\ref{def:spinor-2}).

\smallskip

\noindent (ii) It is easy to see that if $\varphi:\Omega\rightarrow \Omega'$ is a conformal mapping, then we have
\begin{equation}
\label{eq: obs_covariance}
f_{[\Omega,a;a_1\dots,a_k]}(z)=f_{[\varphi(\Omega),\varphi(a);\varphi(a_1)\dots,\varphi(a_k)]}(\varphi(z))\cdot \left(\varphi'(z)\right)^{1/2}.
\end{equation}
We use this covariance property to \emph{define} the continuous spinor in arbitrary simply connected domain $\Omega$, using a conformal map to some smooth bounded $\Omega'$.

\smallskip

\noindent (iii) In Section~\ref{sub:12pts-computations} and~the Appendix we give explicit solutions to (\ref{def:spinor-bdry})--(\ref{def:spinor-2}) in the upper half-plane, thus proving its existence.
\end{rem}

\begin{rem}
 \label{rem:cont-spinor-motivation} The first condition (\ref{def:spinor-bdry}) in Definition~\ref{def:spinor-bounded} is a natural counterpart of (\ref{eq: discrete_bc}). The second condition~(\ref{def:spinor-1}) comes from the following observation: a discrete primitive $H_{[\Omega,a;a_1,\dots,a_k]}$ of the ``differential form'' $\Re\mathfrak{e}\,[F_{[\Omega_\delta,a;a_1,\dots,a_k]}^2dz]$ (which may be defined due to the s-holomorphicity property of the discrete observable, see Remark~\ref{rem:intF^2}) remains bounded from below near the branching points $a_1,\dots,a_k$ as $\delta\to 0$. Thus, we impose the same condition for the scaling limits, which means that $\Re\mathfrak{e}\,[\int f_{\left[\Omega,a;a_1,\dots,a_k\right]}^2dz]$ should behave like $c_j\log|z-a_j|$ for some \emph{negative} $c_j$ as $z\to a_j$, implying~(\ref{def:spinor-1}). The third condition (\ref{def:spinor-2}) which fixes the behavior of the continuous spinor near the ``source'' point $a$ is the most delicate one and will be clarified later on (see Lemma~\ref{lem: def_f_C} and Section~\ref{sub:full-plane}, particularly Lemma~\ref{lem: remove_sing}). Note that it is sufficient to assume that $f_{\left[\Omega,a;a_1,\dots,a_k\right]}$ does not blow up faster than $1/\sqrt{z-a}$ at $a$. Indeed, in this case the argument similar to (\ref{(f1-f2)^2-int}) shows that $\lim\limits_{z\rightarrow a}\sqrt{z-a}\cdot f_{[\Omega,a;a_1,\dots,a_k]}(z)>0$ and the rest is just the proper choice of normalization.
\end{rem}

Let us now introduce the quantities that play a central role in the computation of scaling limits of spin-spin correlations, appearing as the limits of their discrete logarithmic derivatives in Theorem~\ref{thm:log-derivatives}.
\begin{defn}
We define the complex number $\mathcal{A}_\Omega(a;a_1,\dots,a_k)=\mathcal{A}_{[\Omega,a;a_1,\dots,a_k]}$ as the coefficient in the asymptotic expansions
\begin{equation}
\label{eq: expansion-a}
f_{[\Omega,a;a_1,\dots,a_k]}=\frac{1}{\sqrt{z-a}}+2\mathcal{A}_{[\Omega,a;a_1,\dots,a_k]}\sqrt{z-a} + O((z-a)^{3/2})
\end{equation}
near the point $a$. For the special case $k=1$, we also define the quantity $\mathcal{B}_\Omega(a;b)=\mathcal{B}_{[\Omega,a;b]}>0$ as the coefficient in the asymptotic expansion of the spinor $f_{[\Omega,a,a_1,\dots,a_k]}$ near $b$:
\begin{equation}
\label{eq: expansion-b}
f_{[\Omega,a;b]}=\pm \frac{i\mathcal{B}_{[\Omega,a;b]}}{\sqrt{z-b}} + O(\sqrt{z-b}\,),
\end{equation}
where the sign $\pm$ depends on the sheet of $[\Omega,a;b]$.
\end{defn}
\begin{rem}
Note that the covariance rule (\ref{A-covariance}) for $\mathcal{A}_{[\Omega,a;a_1,\dots,a_k]}$ directly follows from the conformal covariance (\ref{eq: obs_covariance}) of spinor observables: if $\varphi:\Omega\to\Omega'$ is a conformal mapping, then one has
\begin{eqnarray*}
f_{\left[\Omega,a;a_1,\dots,a_k\right]}\left(z\right) & = & (\varphi'\left(z\right))^{1/2}f_{\left[\varphi\left(\Omega\right),\varphi\left(a\right);\varphi(a_1),\dots,\varphi(a_k)\right]}(\varphi(z))
 ~=~ \left[\frac{\varphi'\left(z\right)}{\varphi\left(z\right)-\varphi\left(a\right)}\right]^\frac{1}{2} \cdot\left[1+2\mathcal{A}_{\varphi}\cdot(\varphi\left(z\right)-\varphi\left(a\right))+\dots\right]\\
 & = &
 \biggl[\frac{1+\frac{\varphi''(a)}{\varphi'(a)}(z-a)+\dots}{(z-a)(1+\frac{\varphi''(a)}{2\varphi'(a)}(z-a)+\dots)}\biggr]^\frac{1}{2} \cdot \left[1+2\mathcal{A}_{\varphi}\cdot\varphi'(a)(z-a)+\dots\right] \\
 & = &
\frac{1}{\sqrt{z-a}}\left[1+2\left(\mathcal{A}_\varphi\cdot\varphi'\left(a\right)+\frac{1}{8}\frac{\varphi''\left(a\right)}{\varphi'\left(a\right)}\right)(z-a)+\dots\right],
\end{eqnarray*}
where $\mathcal{A}_{\varphi}=\mathcal{A}_{\left[\Omega',\varphi\left(a\right);\varphi(a_1),\dots,\varphi(a_k)\right]}$. Similar arguments show that the coefficient $\mathcal{B}_{[\Omega,a;b]}$ is conformally invariant, i.e.,
\begin{equation}
\label{B-invariance}
\mathcal{B}_\Omega(a;b)=\mathcal{B}_{\Omega'}(\varphi(a);\varphi(b)).
\end{equation}
Thus, it is sufficient to find those quantities for some canonical domain, e.g., for the upper half-plane $\mathbb{H}$. This is done in Section~\ref{sub:12pts-computations} for $k=0,1$ and in the Appendix for $k\ge 2$.
\end{rem}

\subsection{Convergence of spinors}\label{sub:convergence-of-spinors}
The main purpose of this section is to give a rough sketch of the proofs of Theorem~\ref{thm:log-derivatives} and Theorem~\ref{thm:ratio-free-plus}. The complete proofs are given in Section~\ref{sec:analysis-of-spinors} below, which is the most technical part of our paper. We use the following conventions concerning convergence of discrete s-holomorphic functions:
\begin{itemize}
\item we say that a family of discrete domains $\left(\Omega_{\delta}\right)_{\delta}$ approximates a continuous domain $\Omega\subset\mathbb{C}$ as $\delta\to 0$, if $\partial\Omega_{\delta}$ converges to $\partial\Omega$ in the Hausdorff sense (note that our proofs can be easily generalized for the Carath\'eodory convergence of planar domains which is weaker than the Hausdorff convergence used in this paper for simplicity);
\item we say that an s-holomorphic functions (or a spinors) $F_\delta$ defined in $\Omega_\delta^\mathrm{cm}$ (or its double cover) tend to a holomorphic function (or a spinor) $f$ as $\delta\to 0$, if the ``mid-edge values'' $F_\delta\big|_{\Omega_\delta^\mathrm{m}}$ approximate the values of $f$, while the ``corner values'' $F\big|_{\Omega_\delta^\tau}$, $\tau\in\{1,\lambda,i,\overline{\lambda}\}$, tend to the projections of $f$ onto the corresponding lines $\tau\mathbb{R}$ (see Definition~\ref{def: shol});
\item we say that a convergence of discrete functions $F_\delta(z)=F_\delta(z;a,b,\dots)$ to $f(z)=f(z;a,b,\dots)$ is uniform on some compact set, iff the differences $|F_\delta(z;a,b,\dots)-f(z;a,b,\dots)|$ are uniformly small as $\delta\to 0$, when we interpret lattice vertices (or mid-edges, corners, etc) $z,a,b,\dots$  as the corresponding complex points when we plug them into $f$.
\end{itemize}

The crucial ingredient of our proofs is the interplay between (a) the values of discrete spinor observables near their branching points, which are related to the ratios of spin correlations by Lemma~\ref{lem:ratios-disc-magnet-spinor}, and (b) the mid-range behavior of these observables, which can be further related to the asymptotics expansions of their scaling limits (\ref{eq: expansion-a}),~(\ref{eq: expansion-b}).

As a main tool to relate (a) and (b), we use a \emph{full-plane version $F_{\left[\mathbb{C}_{\delta},a\right]}$ of the spinor observable} (since we are interested in local considerations, it is sufficient to stick to the case of one marked point). In principle, it can be constructed as an infinite-volume limit of the finite-domain observables, but we prefer a more descriptive strategy, which is outlined after the following lemma claiming the existence of $F_{\left[\mathbb{C}_{\delta},a\right]}$.

\begin{lem}
\label{lem: def_f_C}
For $a\in\mathcal{V}_{\mathbb{\C}_\delta}^\circ$, there exists a (unique) s-holomorphic spinor $F_{\left[\mathbb{C}_{\delta},a\right]}:\mathcal{V}^{cm}_{[\C,a^\rightarrow]}\rightarrow \C$ such that $F_{\left[\mathbb{C}_{\delta},a\right]}(a+\frac{3\delta}{2})=1$ and $F(z)=o(1)$ as $z\to\infty$. Moreover,
\begin{eqnarray}
\label{F_C-convergence}
\frac{1}{\vartheta(\delta)}F_{\left[\mathbb{C}_{\delta},a\right]}(z) & \underset{\delta\to0}{\longrightarrow} &  \frac{1}{\sqrt{z-a}}~=:f_{[\mathbb{C},a]}(z),
\end{eqnarray}
uniformly on compact subsets of $\mathbb{C}\setminus \{a\}$, where the normalizing factor $\vartheta(\delta)$ is defined as
\begin{equation}
\label{vartheta-def}
\vartheta(\delta):=F_{[\C_\delta,a]}\left(a+\tfrac{3\delta}{2}+2\delta\lfloor\tfrac{1}{2\delta}\rfloor\right).
\end{equation}
\end{lem}

\begin{proof} The detailed proof is given in Section~\ref{sub:full-plane}. First, we define the (real) values of $F_{[\C_\delta,a]}$ on ${\mathcal{V}_{\mathbb{\C}_\delta}^1}$ as a discrete harmonic measure of the tip point $a+\frac{3\delta}{2}$ in the slit discrete plane ${\mathcal{V}_{\mathbb{\C}_\delta}^1}\setminus\{a+x,x\le 0\}$. This definition is motivated by the following observation: in the continuous setup, the function $\Re\mathfrak{e}\,[{1}/{\sqrt{z-a}}\,]$ can be viewed as the properly normalized harmonic measure of the (small neighborhood of) tip $a$ in the slit plane $\mathbb{C}\setminus\{a+x,x\le 0\}$. Second, we extend $F_{[\C_\delta,a]}$ to $\mathcal{V}_{\mathbb{\C}_\delta}^i$ by harmonic conjugation, then by symmetry to another sheet of $[\C_\delta,a]$, and eventually as an s-holomorphic function to $\mathcal{V}^{\lambda}_{\mathbb{C}_\delta},\mathcal{V}^{\overline{\lambda}}_{\mathbb{C}_\delta}$ and $\mathcal{V}^\mathrm{m}_{\mathbb{C}_\delta}$. The convergence~(\ref{F_C-convergence}) follows from known results on convergence of the harmonic measures~\cite{chelkak-smirnov-i}.
\end{proof}

\begin{rem}
The normalizing factor $\vartheta(\delta)$ is essentially the value of $F_{[\C_\delta,a]}$ at $a+1$. In Section~\ref{sub:full-plane} we show that
\begin{equation}
\label{eq: vartheta-estimate}
\mathrm{C}_{-}\sqrt{\delta}\leq\vartheta\left(\delta\right)\leq\mathrm{C}_{+}\sqrt{\delta},
\end{equation}
where $C_\pm>0$ are some absolute constants. Note that one can compute the limit $\lim_{\delta\to 0}\vartheta(\delta)/\sqrt{\delta}$ using the recent work of Dub\'edat \cite{dubedat-i}, but we do not need this sharp result.
\end{rem}

We further use the normalizing factors $\vartheta(\delta)$ introduced above in order to formulate the following convergence theorem for discrete spinor observables away from $\partial\Omega$ and $a,a_1,\dots,a_k$:
\begin{thm}
\label{thm:obs-away}
Let discrete simply connected domains $\Omega_{\delta}$ approximate a bounded simply connected domain $\Omega$ as $\delta\to 0$. Then, for any $\epsilon>0$ and any $k=0,1,\dots$, we have
\begin{eqnarray*}
\frac{1}{\vartheta\left(\delta\right)}F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\right]}(z) & \underset{\delta\to0}{\longrightarrow} & f_{\left[\Omega,a;a_1,\dots,a_k\right]}(z),
\end{eqnarray*}
uniformly over all $a,a_1,\dots,a_k\in\mathcal{V}_{\Omega_{\delta}}^{\circ}$ and $z\in\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}$ which are at the distance at least $\epsilon$ from $\partial\Omega$ and from each other.
\end{thm}
\begin{proof}
See Section~\ref{sub:convergence-away-sing}.
\end{proof}

Since we are interested in the coefficient $\mathcal{A}_{[\Omega,a;a_1,\dots,a_k]}$ in front of the term $\sqrt{z-a}$ in (\ref{eq: expansion-a}), along with a discrete analogue $F_{[\mathbb{C}_\delta,a]}$ of the function $1/\sqrt{z-a}$ given by Lemma~\ref{lem: def_f_C}, we also need a discrete counterpart $G_{[\mathbb{C}_\delta,a]}$ of the function $\sqrt{z-a}$. We construct $G_{[\mathbb{C}_\delta,a]}$ by the ``discrete integration'' of $F_{[\mathbb{C}_\delta,a]}$, just mimicking the continuous setup. It is sufficient to define $G_{[\mathbb{C}_\delta,a]}$ on $\mathcal{V}_{\mathbb{C}_\delta}^1$ only, as Lemma~\ref{lem:ratios-disc-magnet-spinor} deals with the (real) values of discrete observables at the point $a+\tfrac{3\delta}{2}\in \mathcal{V}_{\mathbb{C}_\delta}^1$.

\begin{lem}
\label{def_r_delta}
For $a\in\mathcal{V}_{\mathbb{\C}_\delta}^\circ$, there exists a (unique) discrete harmonic spinor $G_{[\mathbb{C}_\delta,a]}:\mathcal{V}^{1}_{[\C_\delta;a^\rightarrow]}\to \R$ vanishing on the half-line $\{a+x,x\le 0\}$ such that $G_{[\mathbb{C}_\delta,a]}(z)=O(|z-a|^{1/2})$ as $z\to\infty$, and $G_{[\mathbb{C}_\delta,a]}(a+\tfrac{3\delta}{2})=\delta$. Moreover,
\begin{eqnarray}
\label{G_C-convergence}
\frac{1}{\vartheta(\delta)}G_{[\mathbb{C}_\delta,a]}(z)& \underset{\delta\to0}{\longrightarrow} & \Re\mathfrak{e}\,\sqrt{z-a}~=:g_{[\mathbb{C},a]}(z),
\end{eqnarray}
uniformly on compact subsets of $\mathbb{C}\setminus \{a\}$.
\end{lem}
\begin{proof}
For $z\in\mathcal{V}_{[\mathbb{\C}_\delta,a]}^1$, we set
$G_{[\mathbb{C}_\delta,a]}(z):=\delta\cdot\sum_{j=0}^{\infty} F_{[\mathbb{C}_\delta,a]}(z-2j\delta)$ (certainly, one should check the convergence of this series and the harmonicity on $\{a+x,x>0\}$). See further details in Section~\ref{sub:full-plane}.
\end{proof}

In the continuum limit, the leading term in the expansion of $f_{[\Omega,a;a_1,\dots,a_k]}-f_{[\C,a]}$ near $a$ is given by $2\mathcal{A}_{[\Omega,a;a_1,\dots,a_k]}\sqrt{z-a}$. It is hence plausible to believe that the same holds true for the discrete spinors, and one has
\[
(F_{[\Omega_\delta,a;a_1,\dots,a_k]}-F_{[\C_\delta,a]})\left(a+\tfrac{3\delta}{2}\right) ~\approx~ 2\Re\mathfrak{e}\,\mathcal{A}_{[\Omega_\delta,a;a_1,\dots,a_k]}\cdot G_{[\C_\delta,a]}\left(a+\tfrac{3\delta}{2}\right)
\]
up to higher-order terms (the real part appears due to discrete complex analysis subtleties, as real and imaginary parts of s-holomorphic functions are defined on different lattices, and $a+\frac{3\delta}{2}\in\mathcal{V}^{1}_{\Omega_\delta}$).
We justify this heuristics in
\begin{thm}
Under conditions of Theorem~\ref{thm:obs-away}, we have
\begin{eqnarray}
F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\right]}\left(a+\tfrac{3\delta}{2}\right)-1- 2\Re\mathfrak{e}\,\mathcal{A}_{\left[\Omega,a;a_1,\dots,a_k\right]}\cdot\delta & = & o\left(\delta\right)
\label{eq:localization-near-a}
\end{eqnarray}
as $\delta\to 0$, uniformly over all $a,a_1,\dots,a_k\in\mathcal{V}_{\Omega_{\delta}}^{\circ}$ which are at the distance at least $\epsilon$ from $\partial\Omega$ and from each other.
\label{thm:localization-near-a}
\end{thm}
\begin{proof}
See Section~\ref{sub:convergence-near-sing}.
\end{proof}

\begin{rem}
\label{rem:ratios-spinspin-nearby}
In the proof of Theorem~\ref{thm:localization-near-a} we show that $F_{\left[\Omega_{\delta},a;a_1,\dots,a_k\right]}$ and $F_{\left[\C_\delta,a\right]}$ are $\delta$-close to each other at all points around $a$, in particular at $a+(1\pm\frac{i}{2})\delta$. Together with (\ref{eq:nearby-ratios-disc-magnet-spinor}), this implies
\begin{eqnarray*}
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a+(1\pm i)\delta}\sigma_{a_1}\dots\sigma_{a_k}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\right]}  & \underset{\delta\to0}{\longrightarrow} & 1,
\end{eqnarray*}
uniformly over all $a,a_1,\dots,a_k\in\mathcal{V}_{\Omega_{\delta}}^{\circ}$ which are at the distance at least $\epsilon$ from $\partial\Omega$ and from each other.
\end{rem}

The similar analysis for the quantity $\mathcal{B}_{[\Omega,a;b]}$ which is defined, for $k=1$, by the asymptotic expansion (\ref{eq: expansion-b}), is even simpler since we need to match the first-order coefficients instead of the second-order ones. The result is given by

\begin{thm}
Under conditions of Theorem~\ref{thm:obs-away} for $k=1$, we have
\begin{eqnarray}
F_{\left[\Omega_{\delta},a;b\right]}\left(b+\tfrac{\delta}{2}\right)\pm i\mathcal{B}_{[\Omega,a;b]} & = & o\left(1\right)
\label{eq:localization-near-b}
\end{eqnarray}
as $\delta\to 0$, uniformly over all $a,b\in\mathcal{V}_{\Omega_{\delta}}^{\circ}$ which are at the distance at least $\epsilon$ from $\partial\Omega$ and from each other, where the sign $\pm$ depends on the sheet of $[\Omega,a;b]$.
\label{thm:localization-near-b}
\end{thm}
\begin{proof}
See Section~\ref{sub:convergence-near-sing}.
\end{proof}


\begin{proof}[\textbf{Proof of Theorems~\ref{thm:log-derivatives} and~\ref{thm:ratio-free-plus}}]
Due to Lemma~\ref{lem:ratios-disc-magnet-spinor}, asymptotics (\ref{eq:log-der-x}) is a reformulation of (\ref{eq:localization-near-a}), while Theorem~\ref{thm:ratio-free-plus} is equivalent to Theorem~\ref{thm:localization-near-b} (the sign $\pm$ is fixed due to the positivity of spin-spin correlations). To check (\ref{eq:log-der-y}), we rotate our domain $\Omega$  around $a$ by 90\textdegree{} clockwise. According to conformal covariance rule (\ref{A-covariance}), the coefficient $\mathcal{A}$ multiplies by $i$, so the desired result follows from $\Re\mathfrak{e}\,[i\mathcal{A}]=-\Im\mathfrak{m}\,\mathcal{A}$.
\end{proof}

\subsection{Computations for one and two points}\label{sub:12pts-computations}
Now we illustrate the results obtained in the previous section constructing the continuous spinors solving the boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}) for two simplest cases $k=0$ and $k=1$. We use the upper half-plane $\mathbb{H}$ as a canonical domain. Since it is unbounded, one should impose the additional regularity condition at~$\infty$ ensuring that $f_{[\mathbb{H},a;a_1,\dots,a_k]}=O(|z|^{-1})$ as $z\to\infty$,
so that after a conformal mapping to a bounded domain with smooth boundary the spinor remains bounded.

\subsubsection{The case $k=0$} For a single marked point $a\in\mathbb{H}$ we have
\begin{equation}
\label{eq: 1pt-spinor}
f_{[\mathbb{H},a]}(z)=\frac{(2i\Im\mathfrak{m}\, a)^\frac{1}{2}}{\sqrt{(z-a)(z-\overline{a})}},~~z\in[\mathbb{C},a],
\end{equation}
since this spinor clearly satisfies both conditions (\ref{def:spinor-bdry}) and (\ref{def:spinor-2}). Writing down the asymptotic expansion (\ref{eq: expansion-a}) at $a$, one sees that
\[
\mathcal{A}_\mathbb{H}(a)=-\frac{1}{8i\Im\mathfrak{m}\, a}.
\]
Further, we \emph{define} the quantity $\log\langle\sigma_a\rangle_\mathbb{H}^+$ as a primitive of the differential form $\Re\mathrm{e}[\mathcal{A}_\mathbb{H}(a)da]=-{dy}/{8y}$, if $a=x+iy$:
\[
\langle\sigma_a\rangle_\mathbb{H}^+=\mathrm{const}\cdot (\Im\mathfrak{m}\, a)^{-\frac{1}{8}},~~a\in\mathbb{H}.
\]
The proper choice of the multiplicative constant which is discussed below leads us to the definition $\langle\sigma_a\rangle_\mathbb{H}^+:=2^\frac{1}{4}(2\Im\mathfrak{m}\,a)^{-\frac{1}{8}}$.

\subsubsection{The case $k=1$} Similarly, for two marked points $a,b\in\mathbb{H}$, one can check that the spinor
\begin{equation}
\label{eq: 2pts-spinor}
f_{[\mathbb{H},a;b]}(z)=\frac{(2i\Im\mathfrak{m}\, a)^\frac{1}{2}}{|b-\overline{a}|+|b-a|} \cdot \frac{[(\overline{b}-\overline{a})(\overline{b}-a)]^\frac{1}{2}(z-b)+[(b-a)(b-\overline{a})]^\frac{1}{2}(z-\overline{b})} {[(z-a)(z-\overline{a})(z-b)(z-\overline{b})]^{1/2}},~~z\in[\mathbb{C},a;b],
\end{equation}
satisfies (\ref{def:spinor-bdry})--(\ref{def:spinor-2}). Looking at the asymptotic expansions (\ref{eq: expansion-b}) at $b$, one obtains
\[
\mathcal{B}_\mathbb{H}(a;b)=\frac{(4\Im\mathfrak{m}\, a\Im\mathfrak{m}\, b)^\frac{1}{2}}{|b-\overline{a}|+|b-a|}
= \frac{\langle\sigma_a\sigma_b\rangle_\mathbb{H}^\mathrm{free}}{\langle\sigma_a\sigma_b\rangle_\mathbb{H}^+}.
\]

Further, asymptotic expansions near $a$ give
\[
\mathcal{A}_\mathbb{H}(a;b)=-\frac{1}{8i\Im\mathfrak{m}\, a} +\frac{|b-\overline{a}|-|b-a|}{4(|b-\overline{a}|+|b-a|)}\left(\frac{1}{b-a}-\frac{1}{\overline{b}-a}\right).
\]

Straightforward computations shows that one can define a primitive $\int^a\Re\mathfrak{e}\,[\mathcal{A}_\mathbb{H}(z;b)dz]$ for $a\in\mathbb{H}$ (here we treat $b$ as a parameter) that coincides with the function $\log\langle\sigma_a\sigma_b\rangle_\mathbb{H}^+$ given by (\ref{2ptFcts}). Note that $\langle\sigma_a\sigma_b\rangle_\mathbb{H}^+$ is uniquely defined (up to a multiplicative constant) by its logarithmic derivative with respect to $a$ and the symmetry $\langle\sigma_a\sigma_b\rangle_\mathbb{H}^+=\langle\sigma_b\sigma_a\rangle_\mathbb{H}^+$. The multiplicative constants in (\ref{1ptFct}),~(\ref{2ptFcts}) are chosen so that $\langle\sigma_a\sigma_b\rangle_\mathbb{H}^+\sim \langle\sigma_a\rangle_\mathbb{H}^+\langle\sigma_b\rangle_\mathbb{H}^+$ if, say, $a\to \partial\mathbb{H}$, resembling decorrelation properties of the critical Ising model which are given in Section~\ref{sub:proof-thm-k-pts}. Below we show that, for all $k$, the similar integration procedure
allows one to \emph{define} real symmetric functions $\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_\mathbb{H}^+$ such that
\begin{equation}
\label{conf-decorrelation}
\langle \sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+} \sim \langle \sigma_{a}\rangle_{\Omega}^{+} \langle\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+},\quad a\to\partial\Omega.
\end{equation}

\subsection{Proof of Corollary~\ref{cor:conv-up-to-normalization} and decorrelation identities}
\label{sub:decorrelations}
Denote $a_0:=a$. The main purpose of this section is to define the functions $\langle\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ basing on their logarithmic derivatives with respect to each of $a_j$ and (eventual) decorrelation properties (\ref{conf-decorrelation}). Let us denote $a_j=x_j+iy_j$, and consider a differential form $\mathcal{L}_{\Omega,k+1}$ on the manifold $\widetilde{\Omega}^{k+1}:=\{(a_0,\dots a_k)\in\Omega^{k+1}:a_j\neq a_m, j<m\}$ of all $(k+1)$-tuples of distinct points in $\Omega$, defined as follows:
\begin{equation}
\label{eq: L-def}
\mathcal{L}_{\Omega,k+1}:=\sum\limits_{j=0}^k\Re\mathfrak{e}\, \left[\mathcal{A}_\Omega(a_j;a_1,..,\hat{a}_j,..,a_k)da_j\right]=\sum\limits_{j=0}^k \left[\Re\mathfrak{e}\, \mathcal{A}_\Omega(a_j;a_1,..,\hat{a}_j,..,a_k)dx_j-\Im\mathfrak{m}\, \mathcal{A}_\Omega(a_j;a_1,..,\hat{a}_j,..,a_k)dy_j\right]
\end{equation}
(we use the standard notation $\hat{a}_j$ for the omitted argument). The following proposition is a straightforward corollary of Theorem~\ref{thm:log-derivatives} obtained via integration with respect to positions of points:
\begin{prop}
The form $\mathcal{L}_{\Omega,k+1}$ is exact. If $\langle\sigma_{a_0}\dots\sigma_{a_k}\rangle^{+}_{\Omega}:=\exp[\int\mathcal{L}_{\Omega,k+1}]$ denotes the exponential of its primitive, then, as $\Omega_\delta$ approximates $\Omega$, one has
\begin{equation}
\label{eq: conv_ratios}
 \frac{\E^{+}_{\Omega_{\delta}}[\sigma_{b_0}\sigma_{b_1}\dots\sigma_{b_k}]}{\E^{+}_{\Omega_{\delta}}[\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}]}
~\underset{\delta\to0}{\longrightarrow}~
\frac{\langle\sigma_{b_0}\sigma_{b_1}\dots\sigma_{b_k}\rangle^{+}_{\Omega}}{\langle\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle^{+}_{\Omega}}
\end{equation}
for all $(k+1)$-tuples $\{a_j\}_0^k,\{b_j\}_0^k\in\widetilde{\Omega}^{k+1}$.
\label{prop: conv_ratios}
\end{prop}
\begin{rem*}
Note that (\ref{eq: conv_ratios}) is essentially equivalent to the claim of Corollary~\ref{cor:conv-up-to-normalization}, and the proof also ensures that this convergence is uniform, if all $a_j$ are at distance at least $\epsilon$ from $\partial\Omega$ and each other, and the same holds true for $b_j$.
\end{rem*}


\begin{proof}
Color the faces of $\Omega_{\delta}$ black and white, in a chessboard
fashion. By Remark \ref{rem:ratios-spinspin-nearby}, the ratio of spin-spin correlations
at two adjacent spins tends to $1$, uniformly away from the boundary,
so we can assume that all $a_j,b_j$ are colored white. Let $a'_0\in\Omega$ be such that $[a_0,a'_0]$ is a horizontal segment contained in $\Omega$ and disjoint with $a_1,\dots,a_k$. Denote by \mbox{$a_0=v_{0},v_{1},\dots,v_{n_{\delta}}=a_0'$} a straight horizontal lattice path approximating $[a_0,a'_0]$. Then, by Theorem \ref{thm:log-derivatives}, one has
\[
\log\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{v_{j+1}}\sigma_{a_1}\dots\sigma_{a_k}\right]}{\mathbb{E}_{\Omega_{\delta}^{*}}^{+}\left[\sigma_{v_{j}}\sigma_{a_1}\dots\sigma_{a_k}\right]} =\left(\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{v_{j+1}}\sigma_{a_1}\dots\sigma_{a_k}\right]}{\mathbb{E}_{\Omega_{\delta}^{*}}^{+}\left[\sigma_{v_{j}}\sigma_{a_1}\dots\sigma_{a_k}\right]}-1\right)(1 +o(1))=2\delta\cdot[\Re\mathfrak{e}\,\mathcal{A}_\Omega(v_j;a_1,\dots,a_k)+o(1)]
\]
as $\delta\to 0$, where the $o(1)$ terms are unform in $j$. Consequently,
\[
\log\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a_0'}\sigma_{a_1}\dots\sigma_{a_k}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\right]}= 2\delta\cdot\sum\limits_{j=1}^{n_\delta}\Re\mathfrak{e}\,\mathcal{A}_\Omega(v_j;a_1,\dots,a_k)+o(1) ~\underset{\delta\to0}{\longrightarrow}~ \int_{[a_0,a_0']}\Re\mathfrak{e}\,\mathcal{A}_\Omega(x+iy;a_1,\dots,a_k)dx.
\]
A similar formula with $-\Im\mathfrak{m}\,\mathcal{A}_\Omega(x+iy;a_1,\dots,a_k)$ in the right-hand side applies to the case when $[a_0,a'_0]$ is a vertical segment. Moreover, one can move other points $a_1,\dots,a_k$ along horizontal and vertical segments as well. Hence, one has
$$
\log \frac{\E^{+}_{\Omega_{\delta}}[\sigma_{b_0}\sigma_{b_1}\dots\sigma_{b_k}]}{\E^{+}_{\Omega_{\delta}}[\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}]} ~\underset{\delta\to0}{\longrightarrow}~ \int_\gamma\mathcal{L}_{\Omega,k},
$$
where $\gamma$ is any path in $\widetilde{\Omega}^{k+1}$ that connects $(k+1)$-tuples $\{a_j\}_0^k$ and $\{b_j\}_0^k$ and consists of segments (in $\widetilde{\Omega}^{k+1}$) with all but one coordinates fixed. Since the left-hand side does not depend on the choice of the path, the form $\mathcal{L}_{\Omega,k+1}$ is exact.
\end{proof}
\begin{rem}
\label{rem: covariance-ratios}
Note that $\langle\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ is a symmetric function of $a_0,a_1,\dots,a_k$ since it is constructed as a primitive of the symmetric differential form $\mathcal{L}_{\Omega,k+1}$.
 Integrating the conformal covariance rule (\ref{A-covariance}) for $\mathcal{A}_\Omega$, one immediately obtains
\begin{eqnarray*}
\log\frac{\langle\sigma_{b_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}}{\langle\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}} & = & \int_{a_0}^{b_0}\Re\mathfrak{e}\,[\mathcal{A}_\Omega(z;\Ak)dz] \\
&= & \int_{a_0}^{b_0}\Re\mathfrak{e}\,\left[\mathcal{A}_{\varphi(\Omega)}(\varphi(z);\varphi(a_1),\dots,\varphi(a_k))\varphi'(z)dz\right]+ \frac{1}{8}\int_{a_0}^{a_0'}\Re\mathfrak{e}\,\left[\frac{\varphi''(z)}{\varphi'(z)}dz\right] \\
\\
& = & \log\frac{\langle\sigma_{\varphi(b_0)}\sigma_{\varphi(a_1)}\dots\sigma_{\varphi(a_k)}\rangle_{\varphi(\Omega)}^{+}} {\langle\sigma_{\varphi(a_0)}\sigma_{\varphi(a_1)}\dots\sigma_{\varphi(a_k)}\rangle_{\varphi(\Omega)}^{+}} + \frac{1}{8}\log \frac{|\varphi'(b_0)|}{|\varphi'(a_0)|}
\end{eqnarray*}
Iterating, one arrives at the following covariance rule for the ratios of correlations which is a weaker form of (\ref{conf-cov}):
\begin{equation}
\label{eq: covariance-ratios}
\frac{\langle\sigma_{b_0}\sigma_{b_1}\dots\sigma_{b_k}\rangle_{\Omega}^{+}}{\langle\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}}
=
\frac{\langle\sigma_{\varphi(b_0)}\sigma_{\varphi(b_1)}\dots\sigma_{\varphi(b_k)}\rangle_{\varphi(\Omega)}^{+}} {\langle\sigma_{\varphi(a_0)}\sigma_{\varphi(a_1)}\dots\sigma_{\varphi(a_k)}\rangle_{\varphi(\Omega)}^{+}}\cdot \prod\limits_{j=0}^k\frac{|\varphi'(b_j)|^\frac18}{|\varphi'(a_j)|^\frac18}.
\end{equation}
\end{rem}

Proposition \ref{prop: conv_ratios} defines the continuous correlation functions $\langle\sigma_{a_0}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ up to multiplicative constants, which may depend on $\Omega$ and $k$, since the primitive of $\mathcal{L}_{\Omega,k+1}$ is defined up to an additive constant. A natural way to choose these constants coherently for all domains and any number of points is suggested by the following lemma. Denote by
\begin{equation}
\label{eq: D-def}
\mathcal{D}_\Omega(a;b):=\frac{|a-b|}{\text{dist}(\{a;b\},\partial\Omega)}
\end{equation}
the quantity that measures how deeply in the bulk of $\Omega$ the points $a,b$ are.
Denote also $\langle \sigma_a\sigma_b\rangle_{\C}^{+}:=|a-b|^{-\frac{1}{4}}$.
\begin{lem}
\label{lem: norm_cont_corr}
There exists a unique way to choose the multiplicative normalization of $\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ so that for all domains $\Omega$ and for all $k=0,1,\dots$ the following holds:
\begin{eqnarray}
\label{eq: bdry_decor}
\frac{\langle\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}}{\langle\sigma_a\rangle_{\Omega}^{+}\langle\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}}\to 1 \quad\text{as} \quad a\to\partial\Omega\qquad\text{and}\qquad  \frac{\left\langle \sigma_{a}\sigma_{b}\right\rangle_{\Omega}^{+}}{\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\C}^{+}}\to 1 \quad \text{as}\quad \mathcal{D}_\Omega(a;b)\to 0.
\end{eqnarray}
Being normalized in this way, the correlation functions $\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ satisfy the conformal covariance rule (\ref{conf-cov}).
\end{lem}
\begin{proof}
See Section~\ref{sub:decorrelations-continuous}.
\end{proof}

Recall that the properties (\ref{eq: bdry_decor}) are directly motivated by the corresponding properties of the discrete correlations, see Lemmas~\ref{lem: decor_disc} and~\ref{lem: dec_bdry_discrete}. Summarizing, we \textbf{define} the continuous correlation functions $\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ to be the exponentials of the primitives of $\mathcal{L}_{\Omega,k+1}$, normalized as in Lemma \ref{lem: norm_cont_corr}. As it was discussed in Section~\ref{sub:12pts-computations}, in the particular cases $k=0$ and $k=1$, this definition reproduces formulae (\ref{1ptFct}) and (\ref{2ptFcts}), respectively. The case $k\ge 2$ is discussed in the Appendix.


\subsection{From ratios of correlations to Theorem \ref{thm:2pts}}\label{sub:ratios-to-thm-2pts}
This section is devoted to the proof of Theorem~\ref{thm:2pts}. Our goal is to relate the normalizing factors $\rho_2(\delta,\Omega_\delta)$ from Corollary~\ref{cor:conv-up-to-normalization} (which, in principle, might depend on $\Omega$), with the full-plane normalization $\rho(\delta)$. The proof is based on Theorem~\ref{thm:ratio-free-plus}, which claims the convergence of the ratios of free and $+$ spin-spin correlations to an explicit limit $\mathcal{B}_\Omega(a;b)={\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{\mathrm{free}}}/{\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{+}}$. We also use classical FKG (e.g., see \cite[Chapter 2]{grimmett}) and GHS (see~\cite{griffith-hurst-sherman}) inequalities for the Ising model. The small additional ingredient is given by
\begin{rem}
\label{rem: ratio_free_plus}
The following is fulfilled:
\begin{equation}
\label{eq: ratio_free_plus}
\mathcal{B}_\Omega(a;b)\to 1 ~~\text{as}~~\mathcal{D}_\Omega(a;b)\to 0, \quad \text{and} \quad
\mathcal{B}_\Omega(a;b)\to 0 ~~\text{as}~~ a\to\partial \Omega,
\end{equation}
where the quantity $\mathcal{D}_\Omega(a;b)$ is given by (\ref{eq: D-def}). This follows readily from the conformal invariance of $\mathcal{B}_\Omega(a;b)$ and the explicit formulae (\ref{2ptFcts}) in the half-plane. Also, one has
\begin{equation}
\label{eq: bulk_limit}
\frac{\left\langle \sigma_{a}\sigma_{b}\right\rangle_{\Omega}^{+}}{\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\C}^{+}}\to 1 ~~\text{as}~~ \mathcal{D}_\Omega(a;b)\to 0.
\end{equation}
Indeed, let $\varphi$ be a conformal map from $\Omega$ to $\bH$ such that $\varphi(a)=i$. By standard estimates for conformal maps, when $\mathcal{D}_\Omega(a;b)\to 0$, one has $\varphi(b)\to i$ and $|\varphi'(b)|\to |\varphi'(a)|$. Hence, the explicit formula (\ref{2ptFcts}) imply
\[
\frac{\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\Omega}^{+}}{\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\C}^{+}}
=
\frac{\left\langle \sigma_{\varphi(a)}\sigma_{\varphi(b)}\right\rangle _{\bH}^{+}|\varphi'(a)|^{\frac{1}{8}}|\varphi'(b)|^{\frac{1}{8}}}{\left\langle \sigma_{a}\sigma_{b}\right\rangle _{\C}^{+}}
=
\frac{|\varphi(a)-\varphi(b)|^{-\frac14}|\varphi'(a)|^{\frac{1}{4}}}{|a-b|^{-\frac14}}\left(1+o(1)\right)\to 1.
\]
\end{rem}

\begin{lem}
For any $\eta>0$ there exists an $\epsilon>0$ such that the following holds: if $\mathcal{D}_\Omega(a;b)<\epsilon$ and $\Omega_\delta$ approximates $\Omega$, then
\[
1-\eta \leq\frac{\E_{\C_\delta}[\sigma_{a}\sigma_{b}]}{\E_{\Omega_\delta}^{+}[\sigma_{a}\sigma_{b}]}\leq 1
\]
provided that $\delta$ is small enough.
\label{lem: decor_disc}
\end{lem}
\begin{proof}
By FKG inequality,
$
\mathbb{E}_{\Lambda_{\delta}}^{\mathrm{free}}\left[\sigma_{a}\sigma_{b}\right]\leq\mathbb{E}_{\mathbb{C}_{\delta}}\left[\sigma_{a}\sigma_{b}\right]\leq\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]
$ for any domain $\Lambda_\delta$ containing $a,b$, hence the right-hand side readily follows. For the left-hand side, choose $\Lambda_\delta=\Omega^{\bullet}_\delta-\delta$. We have, by Theorem \ref{thm:ratio-free-plus},
\[
\frac{\mathbb{E}_{\Lambda_{\delta}}^{\mathrm{free}}\left[\sigma_{a}\sigma_{b}\right]}{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}
=
\frac{\mathbb{E}_{\Omega^{\bullet}_\delta}^{\mathrm{free}}\left[\sigma_{a+\delta}\sigma_{b+\delta}\right]} {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]} ~\underset{\delta\to0}{\longrightarrow}~\mathcal{B}_\Omega(a;b).
\]
Due to Remark~\ref{eq: ratio_free_plus}, one can choose $\epsilon$ so that $\mathcal{B}_\Omega(a;b)\ge 1-\frac{\eta}{2}$, which gives the result.
\end{proof}

\begin{proof}[\textbf{Proof of Theorem \ref{thm:2pts}}]
Fix $\eta>0$. For shortness, below we will write $a+\epsilon$ for its lattice approximations $a+2\delta\lfloor\frac{\epsilon}{2\delta}\rfloor$, and $a+1$ for its lattice approximation $a+2\delta\lfloor\frac{1}{2\delta}\rfloor$. Note that
\[
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}{\varrho(\delta)}=
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}{\mathbb{E}_{\mathbb{C}_{\delta}}\left[\sigma_{a}\sigma_{a+1}\right]}=
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}{\mathbb{E}^{+}_{\Omega_{\delta}}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}\cdot
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}{\mathbb{E}_{\mathbb{C}_{\delta}}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}\cdot
\frac{\mathbb{E}_{\C_{\delta}}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}{\mathbb{E}_{\mathbb{C}_{\delta}}\left[\sigma_{a}\sigma_{a+1}\right]}.
\]
By Lemma \ref{lem: decor_disc}, we can find a small $\epsilon>0$ and a large domain $\Lambda_\delta$ containing $a$, $b$, $a+\epsilon$ and $a+1$, such that
\[
1-\eta ~\leq~ \frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}{\mathbb{E}_{\mathbb{C}_{\delta}}\left[\sigma_{a}\sigma_{a+\epsilon}\right]} ~\leq~ 1\quad \text{and}\quad 1-\eta ~\le~ \frac{\mathbb{E}_{\C_{\delta}}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}{\mathbb{E}_{\mathbb{C}_{\delta}}\left[\sigma_{a}\sigma_{a+1}\right]}\cdot
\frac{\mathbb{E}_{\Lambda_{\delta}}^{+}\left[\sigma_{a}\sigma_{a+1}\right]}{\mathbb{E}_{\Lambda_{\delta}}^+\left[\sigma_{a}\sigma_{a+\epsilon}\right]} ~\le~ 1+\eta,
\]
provided that $\delta$ is small enough. Consequently,
\[
(1-\eta)^2
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}{\mathbb{E}^+_{\Omega_{\delta}}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}\cdot
\frac{\mathbb{E}_{\Lambda_{\delta}}^{+}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}{\mathbb{E}^+_{\Lambda_{\delta}}\left[\sigma_{a}\sigma_{a+1}\right]}
~\leq~
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}{\varrho(\delta)}
~\leq~
(1+\eta)
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}{\mathbb{E}^+_{\Omega_{\delta}}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}\cdot
\frac{\mathbb{E}_{\Lambda_{\delta}}^{+}\left[\sigma_{a}\sigma_{a+\epsilon}\right]}{\mathbb{E}^+_{\Lambda_{\delta}}\left[\sigma_{a}\sigma_{a+1}\right]},
\]
and, by convergence of the ratios proven in Proposition~\ref{prop: conv_ratios},
\[(1-\eta)^3
\frac{\langle\sigma_{a}\sigma_{b}\rangle_{\Omega}^{+}}{\langle\sigma_{a}\sigma_{a+\epsilon}\rangle^+_{\Omega}}\cdot
\frac{\langle\sigma_{a}\sigma_{a+\epsilon}\rangle_{\Lambda}^{+}}{\langle\sigma_{a}\sigma_{a+1}\rangle^+_{\Lambda}}
~\leq~
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}{\varrho(\delta)}
~\leq~
(1+\eta)^2
\frac{\langle\sigma_{a}\sigma_{b}\rangle_{\Omega}^{+}}{\langle\sigma_{a}\sigma_{a+\epsilon}\rangle^+_{\Omega}}\cdot
\frac{\langle\sigma_{a}\sigma_{a+\epsilon}\rangle_{\Lambda}^{+}}{\langle\sigma_{a}\sigma_{a+1}\rangle^+_{\Lambda}}
\]
for $\delta$ small enough. Since $\eta$ can be chosen arbitrary small, and the bounds do not depend on $\delta$, it only remains to show that we can make the factor
\[
\frac{\langle\sigma_{a}\sigma_{a+\epsilon}\rangle_{\Lambda}^{+}}{ \langle\sigma_{a}\sigma_{a+\epsilon}\rangle^+_{\Omega}\langle\sigma_{a}\sigma_{a+1}\rangle^+_{\Lambda}}
\]
as close to one as we wish by choosing $\epsilon$ small enough and $\Lambda$ large enough. However, this follows readily from (\ref{eq: bulk_limit}) if we multiply this factor by $\langle\sigma_{a}\sigma_{a+1}\rangle_{\C}=1$. Thus, ${\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]}\sim {\varrho(\delta)} \langle\sigma_{a}\sigma_{b}\rangle_{\Omega}^{+}$ as $\delta\to 0$.
To derive the asymptotics of two-point correlations for free boundary conditions as $\delta\to 0$, note that Theorem~\ref{thm:ratio-free-plus} implies
\[
{\mathbb{E}_{\Omega_{\delta}^\bullet-\delta}^\mathrm{free}\left[\sigma_a\sigma_b\right]}\sim \mathcal{B}_\Omega(a;b)\cdot {\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]} \sim
{\varrho(\delta)}\cdot\mathcal{B}_\Omega(a;b)\langle\sigma_{a}\sigma_{b}\rangle_{\Omega}^{+}= {\varrho(\delta)}\langle\sigma_{a}\sigma_{b}\rangle_{\Omega}^\mathrm{free}.
\]
The fact that we have $\Omega_{\delta}^\bullet-\delta$ instead of $\Omega_\delta$ plays no role, since they both approximate the same continuous domain  $\Omega$ and the convergence of $(\varrho(\delta))^{-1}\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]$ is independent of the particular choice of lattice approximations.
\end{proof}
\begin{rem}
As a simple byproduct of our analysis, we obtain the rotational invariance of the full-plane correlations recently proven by Pinson \cite{pinson}:
by FKG inequality, for any (large) domain $\Omega_\delta$, one has $\mathbb{E}_{\Omega_{\delta}}^\mathrm{free}\left[\sigma_{a}\sigma_{b}\right]\le \mathbb{E}_{\C_{\delta}}\left[\sigma_{a}\sigma_{b}\right] \le \mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{b}\right]$ and, due to Theorem~\ref{thm:2pts} and (\ref{eq: ratio_free_plus}), both sides have the same asymptotics when $\Omega_\delta$ exhausts $\C_\delta$. Then, (\ref{eq: bulk_limit}) gives the desired result:
$(\rho(\delta))^{-1}\E_{\C_\delta}[\sigma_a\sigma_b]\rightarrow \langle\sigma_{a}\sigma_{b}\rangle_{\C}=|a-b|^{-\frac{1}{4}}$ as $\delta\to 0$.
\end{rem}

\subsection{Decorrelation near the boundary and the proof of Theorem \ref{thm:1-3pts}}\label{sub:proof-thm-k-pts}
This section is devoted to the proof of Theorem~\ref{thm:1-3pts}. Note that it was already proven above in the special case $k=1$, as a part of Theorem~\ref{thm:2pts}. Our goal is to relate the normalizing factors in the Corollary \ref{cor:conv-up-to-normalization} with $\varrho(\delta)$. Below we rely upon decorrelation identities (\ref{eq: bdry_decor}).
\begin{lem}
Given a domain $\Omega$ with marked points $a_1,\dots,a_k$ and a number $\eta>0$, there is an $\epsilon>0$ such that the following holds: if $a\in\Omega$ is $\epsilon$-close to the boundary and $\Omega_\delta$ approximates $\Omega$, then
\begin{equation*}
 1-\eta\leq\frac{\E_{\Omega_\delta}^{+}[\sigma_a]\E_{\Omega_\delta}^{+}[\sigma_{a_1}\dots\sigma_{a_k}]}{\E_{\Omega_\delta}^{+}[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}]}\leq 1.
\end{equation*}
 \label{lem: dec_bdry_discrete}
\end{lem}
\begin{proof}
The upper bound follows readily from FKG inequality. For the lower one, consider first the case $k=1$. A celebrated application (see~\cite{dembo-montanari}) of the GHS inequality reads
$
\E_{\Omega_\delta}^{+}[\sigma_{a}\sigma_{a_1}]-\E_{\Omega_\delta}^{+}[\sigma_{a}]\E_{\Omega_\delta}^{+}[\sigma_{a_1}]\leq\E_{\Omega_\delta}^{\text{free}}[\sigma_{a}\sigma_{a_1}],
$
or equivalently,
\[
1-\frac{\E_{\Omega_\delta}^{\text{free}}[\sigma_{a}\sigma_{a_1}]}{\E_{\Omega_\delta}^{+}[\sigma_{a}\sigma_{a_1}]}
\leq\frac{\E_{\Omega_\delta}^{+}[\sigma_{a}]\E_{\Omega_\delta}^{+}[\sigma_{a_1}]}{\E_{\Omega_\delta}^{+}[\sigma_{a}\sigma_{a_1}]}.
\]
As $\delta\to 0$, the left-hand side converges to $1-\mathcal{B}_\Omega(a;a_1)$,
thus the claim follows from (\ref{eq: ratio_free_plus}).

To prove the result for $k\geq 2$, assume that we have already proven Theorem \ref{thm:1-3pts} for all $k'<k$. Let $\gamma$ be a crosscut (simple path) in $\Omega$ separating $a$ from $a_1,\dots,a_k$, and let $\Omega(a)$ and $\Omega(a_1)$ be the corresponding connected components. Then $\E_{\Omega_\delta(a)}^{+}[\sigma_a]\E_{\Omega_\delta(a_1)}^{+}[\sigma_{a_1}\dots\sigma_{a_k}]$ is equal to the correlation of $\sigma_a,\sigma_{a_1},\dots,\sigma_{a_k}$ in $\Omega_\delta$ conditioned on the event that all spins neighboring $\gamma$ are pluses. Hence, FKG inequality implies
\[
\frac{\E_{\Omega_\delta(a)}^{+}[\sigma_a]\E_{\Omega_\delta(a_1)}^{+}[\sigma_{a_1}\dots\sigma_{a_k}]}{\E_{\Omega_\delta}^{+}[\sigma_a]\E_{\Omega_\delta}^{+}[\sigma_{a_1}\dots\sigma_{a_k}]}\cdot
\frac{\E_{\Omega_\delta}^{+}[\sigma_a]\E_{\Omega_\delta}^{+}[\sigma_{a_1}\dots\sigma_{a_k}]}{\E_{\Omega_\delta}^{+}[\sigma_a\sigma_{a_1}\dots\sigma_{a_k}]}
=
\frac{\E_{\Omega_\delta(a)}^{+}[\sigma_a]\E_{\Omega_\delta(a_1)}^{+}[\sigma_{a_1}\dots\sigma_{a_k}]}{\E_{\Omega_\delta}^{+}[\sigma_a\sigma_{a_1}\dots\sigma_{a_k}]}\geq 1.
\]
By our induction assumption, the first term on the left-hand side converges to
\[
\frac{\langle\sigma_a\rangle_{\Omega(a)}^{+}}{\langle\sigma_{a_1}\rangle_{\Omega}^{+}}\cdot
\frac{\langle\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega(a_1)}^{+}}{\langle\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}},
\]
hence it is sufficient to show that we can make this quantity arbitrary close to $1$ by choosing appropriately $a$ and $\gamma$. We first fix a crosscut $\gamma$ in such a way that $\Omega(a_1)$ would be Carath\'eodory close to $\Omega$ as seen from $a_1$ and then put $a$  deeply inside $\Omega_{a}$, so that $\Omega_{a}$ would be Carath\'eodory close to $\Omega$ as seen from $a$. If two domains are Carath\`eodory close, then the conformal maps from these domains to $\bH$ mapping the pole to $i$ (and say, with positive derivative there) are uniformly close on compacts together with their derivatives. Thus, the lemma follows from continuity of the half-plane functions $\langle\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH}^{+}$ with respect to positions of $\Ak$.
\end{proof}

\begin{proof}[\textbf{Proof of Theorem \ref{thm:1-3pts}}] Recall that the special case $k=1$ is already done as a part of Theorem~\ref{thm:2pts}. We proceed by induction which (together with the proof of Lemma~\ref{lem: dec_bdry_discrete} given above) starts as follows: $T_1\& L_1\Rightarrow T_0\Rightarrow L_2\Rightarrow T_2\Rightarrow L_3\Rightarrow T_3\Rightarrow\dots$ (where $T_k$ and $L_k$ mean the particular cases of Theorem~\ref{thm:1-3pts} and Lemma~\ref{lem: dec_bdry_discrete}, respectively).

Fix $\eta>0$. For any $c\in\Omega$, we can write
\[
\left(\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\right]}{\sqrt{\varrho\left(\delta\right)}}\right)^{2}
=
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\right]\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{c}\right]}{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{c}\right]}\cdot
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{c}\right]}{\varrho\left(\delta\right)}\cdot
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\right]}{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{c}\right]}.
\]
By Lemma \ref{lem: dec_bdry_discrete}, if we choose $c$ close enough to the boundary, then
\[
1-\eta\leq\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\right]\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{c}\right]}{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{c}\right]}\leq 1,
\]
provided that $\delta$ is small enough. Due to Theorem~\ref{thm:2pts} and Proposition~\ref{prop: conv_ratios}, one has
\[
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\sigma_{c}\right]}{\varrho\left(\delta\right)} ~\underset{\delta\to0}{\longrightarrow}~ \langle\sigma_{a}\sigma_{c}\rangle_\Omega^+\quad \text{and} \quad
\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\right]}{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{c}\right]} ~\underset{\delta\to0}{\longrightarrow}~ \frac{\langle \sigma_a \rangle_{\Omega}^{+}}{\langle \sigma_c \rangle_{\Omega}^{+}}.
\]
Consequently,
\[(1-\eta)^3\frac{\langle\sigma_{a}\sigma_{c}\rangle_{\Omega}^{+}}{\langle \sigma_a \rangle_{\Omega}^{+}\langle \sigma_c \rangle_{\Omega}^{+}}\left(\langle \sigma_a \rangle_{\Omega}^{+}\right)^2
\leq
\left(\frac{\mathbb{E}_{\Omega_{\delta}}^{+}\left[\sigma_{a}\right]}{\sqrt{\varrho\left(\delta\right)}}\right)^{2}
\leq
(1+\eta)^2\frac{\langle\sigma_{a}\sigma_{c}\rangle_{\Omega}^{+}}{\langle \sigma_a \rangle_{\Omega}^{+}\langle \sigma_c \rangle_{\Omega}^{+}}\left(\langle \sigma_a \rangle_{\Omega}^{+}\right)^2
\]
provided that $\delta$ is small enough. Since $\eta$ is arbitrary, and we can make the term ${\langle\sigma_{a}\sigma_{c}\rangle_{\Omega}^{+}}/{\langle a \rangle_{\Omega}^{+}\langle c \rangle_{\Omega}^{+}}$ arbitrary close to one (recall that this particular case of (\ref{eq: bdry_decor}) follows from explicit computations in the half-plane given in Section~\ref{sub:12pts-computations}), we finish the proof by remark that positivity of magnetization fixes the sign.

To get the convergence of $\varrho(\delta)^{-\frac{k+1}{2}}\E^{+}_{\Omega}[\sigma_{a}\sigma_{a_1},\dots\sigma_{a_k}]$, proceed by induction and write it as
\[
\frac{\E^{+}_{\Omega}[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}]}{\varrho(\delta)^{\frac{k+1}{2}}}
=
\frac{\E^{+}_{\Omega}[\sigma_{a}\sigma_{a_1}\dots\sigma_{a_k}]}{\E^{+}_{\Omega}[\sigma_{c}\sigma_{a_1}\dots \sigma_{a_k}]}
\cdot\frac{\E^{+}_{\Omega}[\sigma_{c}\sigma_{a_1}\dots\sigma_{a_k}]}{\E^{+}_{\Omega}[\sigma_{c}]\E^{+}_{\Omega}[\sigma_{a_1}\dots\sigma_{a_k}]}
\cdot\frac{{\E^{+}_{\Omega}[\sigma_{c}]\E^{+}_{\Omega}[\sigma_{a_1}\dots\sigma_{a_k}]}}{\sqrt{\varrho(\delta)}\varrho^\frac{k}{2}(\delta)}.
\]
The proof is finished as before: take $c$ close to the boundary, estimate the second term in the right-hand side by Lemma~\ref{lem: dec_bdry_discrete}, then use convergence of the first and the third terms (which follows by Proposition~\ref{prop: conv_ratios} and induction hypothesis, respectively) and asymptotics (\ref{eq: bdry_decor}) for the continuous correlations: ${\langle\sigma_c\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}}\sim {\langle\sigma_c\rangle_{\Omega}^{+}\langle\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}}$ as $c\to\partial\Omega$.
\end{proof}


\section{\label{sec:analysis-of-spinors}Proofs of the main convergence Theorems \ref{thm:obs-away}, \ref{thm:localization-near-a} and \ref{thm:localization-near-b} for spinors}

\subsection{\label{sub:s-holomorphicity-proof}S-holomorphicity of discrete observables}
The notion of s-holomorphicity was essentially introduced by Smirnov in \cite{smirnov-i} and
used for the study of the planar critical Ising model in \cite{smirnov-i, smirnov-ii, chelkak-smirnov-ii, hongler-i, hongler-smirnov-ii, chelkak-izyurov}. Our definitions resemble \cite{hongler-smirnov-ii} and are equivalent to those of \cite{chelkak-smirnov-ii,chelkak-izyurov} after the multiplication by $\sqrt{i}$, see also Section~\ref{sub:discrete-integration} below.

Recall that we associate the line $\ell\left(x\right)=\tau\mathbb{R}$ in the
complex plane with each corner $x\in\mathcal{V}_{\mathbb{C}_{\delta}}^{\tau}$
(for $\tau\in\left\{ 1,i,\lambda,\overline{\lambda}\right\} $),
and denote by $\mathsf{P}_{\ell\left(x\right)}$
the projection onto that line:
\[
\mathsf{P}_{\ell\left(x\right)}\left[w\right]=\tfrac{1}{2}\left(w+\tau^{2}\overline{w}\right),\quad\forall w\in\mathbb{C}.
\]

Definition~\ref{def: shol} says that a function $F:\mathcal{V}_{\Omega_{\delta}}^{\mathrm{cm}}\to\mathbb{C}$
is \emph{s-holomorphic in $\Omega_\delta$} (we also use the same definitions for double covers) if for every $x\in\mathcal{V}_{\Omega_{\delta}}^{\mathrm{c}}$
and $z\in\mathcal{V}_{\Omega_{\delta}}^{\mathrm{m}}$ that are adjacent, one has
\[
F\left(x\right)=\mathsf{P}_{\ell\left(x\right)}\left[F\left(z\right)\right].
\]

\begin{rem}
\label{rem: s-hol-extended}
The set $\mathcal{V}^{1,i}_{\C_\delta}=\mathcal{V}^{1}_{\C_\delta}\cup\mathcal{V}^{i}_{\C_\delta}$ may be viewed as a square lattice, divided into $\mathcal{V}^{1}_{\C_\delta}$, $\mathcal{V}^{i}_{\C_\delta}$ in a chessboard fashion. By definition, the restriction of an s-holomorphic function $F$ to $\mathcal{V}^{1,i}_{\C_\delta}$ is real on $\mathcal{V}^{1}_{\C_\delta}$ and purely imaginary on $\mathcal{V}^{i}_{\C_\delta}$. It is not difficult to check (\cite{smirnov-i,chelkak-smirnov-ii}) that this restriction is in fact \emph{discrete holomorphic}, that is, for any $a\in\mathcal{V}^{1,i}_{\C_\delta}$ one has
\begin{equation}
i\cdot\left[F(a+(1+i)\delta)-F(a)\right]=F(a+i\delta)-F(a+\delta).
\label{eq: disc_holom}
\end{equation}
Moreover, the converse is also true: given a discrete holomorphic function $F:\mathcal{V}^{1,i}_{\C_\delta}\to \R\cup i\R$, we can first extend it to $\mathcal{V}^{m}_{\C_\delta}$ by the formula $F(x):=F(x+i\frac{\delta}{2})+F(x-i\frac{\delta}{2})$ and then to $\mathcal{V}^{\lambda,\overline{\lambda}}_{\C_\delta}$ by $F\left(x\right):=\mathsf{P}_{\ell\left(x\right)}\left[F\left(x\pm\frac{\delta}{2}\right)\right]$, since (\ref{eq: disc_holom}) ensures that the result is the same for both choices.
\end{rem}

We now check the s-holomorphicity of discrete spinor observables, essentially mimicking \cite{chelkak-smirnov-ii,hongler-smirnov-ii,chelkak-izyurov}.
\begin{proof}[\textbf{Proof of Proposition \ref{prop: s-hol+bc}.}]
Let $z\in\mathcal{V}_{\left[\Omega_{\delta},a;\Ak\right]}^{\mathrm{m}}$
be a medial vertex and $x\in\mathcal{V}_{\left[\Omega_{\delta},a;\Ak\right]}^{\mathrm{c}}$
be one of four nearby corners (so that $|x-z|=\frac{\delta}{2}$). We should check that
\begin{eqnarray}
\label{eq: s-hol-spinor}
\mathsf{P}_{\ell\left(x\right)}\left[F_{\left[\Omega_\delta,a;\Ak\right]}\left(z\right)\right] & = & F_{\left[\Omega_\delta,a;\Ak\right]}\left(x\right),
\end{eqnarray}
where both sides are defined as sums over the contour sets $\mathcal{C}_{\Omega_{\delta}}\left(a+\frac{\delta}{2},z\right)$
and $\mathcal{C}_{\Omega_{\delta}}\left(a+\frac{\delta}{2},x\right)$, respectively. There is an obvious bijection $\gamma\mapsto\omega(\gamma)$ between these two sets provided by taking XOR (symmetric difference) of a configuration with two half edges $(zv)$ and $(vx)$, where $v$
denotes the vertex which is adjacent to both $x$ and $z$. Hence, it is sufficient to check that for any configuration $\gamma\in \mathcal{C}_{\Omega_{\delta}}\left(a+\frac{\delta}{2},z\right)$ one has
\begin{equation}
(\cos\tfrac{\pi}{8})^{-1}\cdot\mathsf{P}_{\ell\left(x\right)}\left[\alpha_c^{\#\textrm{edges}(\gamma)}\cdot\phi_{a,\Ak}(\gamma,z)\right]  = \alpha_c^{\#\textrm{edges}(\omega(\gamma))}\cdot\phi_{a,\Ak}(\omega(\gamma),x)
\label{eq: s-hol-reduced}
\end{equation}
(the additional factor $(\cos\frac{\pi}{8})^{-1}$ comes from our defintion of the discrete observable on medial vertices, see Remark~\ref{rem:spinor-def}).
There are two cases: either $(zv)$ is contained in $\gamma$, which leads us to
\begin{equation*}
\#\textrm{edges}(\omega(\gamma))=\#\textrm{edges}(\gamma)\quad\text{and}\quad \exp[-\tfrac{i}{2}\wind(p(\omega(\gamma)))]=e^{\pm\frac{i\pi}{8}}\exp[-\tfrac{i}{2}\wind(p(\gamma))],
\end{equation*}
or not, which gives
\begin{equation*}
\#\textrm{edges}(\omega(\gamma))=\#\textrm{edges}(\gamma)+1\quad\text{and}\quad \exp[-\tfrac{i}{2}\wind(p(\omega(\gamma)))]=e^{\pm\frac{3i\pi}{8}}\exp[-\tfrac{i}{2}\wind(p(\gamma))].
\end{equation*}
Let us also note that in both cases
\[
\left(-1\right)^{\#\mathrm{loops}_{a;\Ak}(\gamma)}\mathrm{sheet}_{a;\Ak}(\gamma,z)=
\left(-1\right)^{\#\mathrm{loops}_{a;\Ak}(\omega(\gamma))}\mathrm{sheet}_{a;\Ak}(\omega(\gamma),z)
\]
since if $\omega$ destroys a loop that changed the sheet of $\omega$ (leading to the change of the first factor), then this loop becomes a part of $\mathrm{p}(\gamma)$, so the second factor changes simultaneously. Thus, one can factor out  $\alpha_c^{\#\textrm{edges}(\omega(\gamma))}\cdot\phi_{a,\Ak}(\omega(\gamma),x)$ from both sides of (\ref{eq: s-hol-reduced}). In the first case (\ref{eq: s-hol-reduced}) readily follows, while in the second one it becomes equivalent to
\[
(\cos\tfrac{\pi}{8})^{-1}\cos\tfrac{3\pi}{8}=\sqrt{2}-1=\alpha_c
\]

Thus, $F_{[\Omega,a;\Ak]}$ is s-holomorphic. The factor $\mathrm{sheet}_{a;\Ak}(\gamma,z)$ in the definition guarantees that it has multiplicative monodromy $-1$ around each of the marked points $a,a_1,\dots,a_k$, and in order to prove that $f_{[\Omega,a,\Ak]}$ obeys the boundary conditions (\ref{eq: discrete_bc}), it is sufficient to note that, if $z$ is on the boundary, then  $\wind(\mathrm{p}(\gamma))=\nu_\mathrm{out}(z) \mod 2\pi$.
\end{proof}

Note that the spinor $F_{[\Omega,a;\Ak]}$ is not s-holomorphic at $a+\frac{\delta}{2}$, as the identities for windings in the above proof fail. The next lemma shows, in particular, that its values at the nearby medial vertices $a+\frac{1\pm i}{2}\delta$ have different projections onto the imaginary line $i\R=\ell(a+\frac{\delta}{2})$, so one cannot extend $F_{[\Omega,a;\Ak]}$ to $a+\frac{\delta}{2}$ in an s-holomorphic way.
\begin{lem}
\label{lem: sing_f_Omega}
For medial vertices $a+\frac{1\pm i}{2}\delta$ taken on the same sheet of $\left[\Omega_\delta,a;\Ak\right]$ as the ``source'' $a+\frac{\delta}{2}$, one has
\[
 \mathsf{P}_{i\mathbb{R}}\left[F_{\left[\Omega_\delta,a;\Ak\right]}\left(a+\tfrac{1\pm i}{2}\delta\right)\right]  =  \mp i,
\]
\end{lem}
\begin{proof}
We consider the point $a+\frac{1+i}{2}$, the other case is similar. Given a configuration $\gamma\in \mathcal{C}_\delta(a+\frac{\delta}{2},a+\delta\frac{1+i}{2})$ and applying, as above, the XOR bijection with two half-edges $(a+\frac{\delta}{2},a+\delta)$ and $(a+\delta,a+\frac{1+i}{2}\delta)$, we obtain a configuration $\omega(\gamma)\in\mathcal{C}_\delta$. Since the normalizing factor $\mathcal{Z}_{\Omega_\delta}^+[a,\Ak]$ is a sum over $\mathcal{C}_\delta$, it is sufficient to show that, for any $\gamma$,
\begin{equation}
(\cos\tfrac{\pi}{8})^{-1}\cdot\mathsf{P}_{i\mathbb{R}}\left[\alpha_c^{\#\textrm{edges}(\gamma)}\phi_{a;\Ak}(\gamma,a+\tfrac{1+i}{2}\delta)\right]= -i(-1)^{\#\text{loops}_{a;\Ak}(\omega(\gamma))}\alpha_c^{\#\text{edges}(\omega(\gamma))}
\label{eq: check_projections}
\end{equation}
Consider two cases, as in the proof above. If $(a+\delta,a+\frac{1+i}{2}\delta)\in\gamma$ (respectively, $(a+\delta,a+\frac{1+i}{2}\delta)\notin\gamma$), then
\[
\#\textrm{edges}(\omega(\gamma))=\#\textrm{edges}(\gamma)\quad\left(\text{respectively,}\quad \#\textrm{edges}(\omega(\gamma))=\#\textrm{edges}(\gamma)+1\right).
\]
Also, we may disregard all loops in $\omega(\gamma)$ that do not contain the edge $(a+\delta,a+i\delta)$, as they contribute the same sign to both sides. If $(a+\delta,a+\frac{1+i}{2}\delta)\in\gamma$, then $\wind(\mathrm{p}(\gamma))=\frac{3\pi}{4}\mod 4\pi$, otherwise $\wind(\mathrm{p}(\gamma))=\frac{7\pi}{4}\mod 4\pi$. Hence the lemma boils down to the following elementary identities:
\[
(\cos\tfrac{\pi}{8})^{-1}\cdot \mathsf{P}_{i\mathbb{R}}[e^{-\frac{3\pi i}{8}}]=-i\quad\text{and}\quad
(\cos\tfrac{\pi}{8})^{-1}\cdot \mathsf{P}_{i\mathbb{R}}[e^{-\frac{7\pi i}{8}}]=-i\alpha_c. \qedhere
\]
\end{proof}

\subsection{The full-plane discrete spinor $F_{[\C_\delta,a^\rightarrow]}$ and its discrete primitive $G_{[\C_\delta,a^\rightarrow]}$} \label{sub:full-plane}
This subsection is mainly devoted to the construction of the full-plane analogue of discrete spinor observables, as announced in Lemma~\ref{lem: def_f_C}. After this, we also prove Lemma~\ref{def_r_delta} and the double-sided estimate (\ref{eq: vartheta-estimate}) of the normalizing factor $\vartheta(\delta)$.

\subsubsection{Discrete harmonic measure in the slitted plane}
We start with an important technical ingredient -- the discrete Beurling estimate with optimal exponent $\frac{1}{2}$.
On the square lattice, it was obtained by Kesten \cite{kesten}, and then generalized by Lawler and Limic \cite{lawler-limic}.
Let $\Xi_\delta$ denote the slitted discrete plane $\mathcal{V}^{1}_{\C_\delta}\backslash L_a$, where $L_a:=\{a+x,x\le \frac{\delta}{2}\}$.

\begin{lem}
\label{lem:discrete-beurling}
For all $z\in\Xi_\delta$ and $A\subset\Xi_\delta$ and some absolute constant $C>0$, the following estimates are fulfilled:
\begin{equation}
\label{eq: hm-beurling}
\mathrm{hm}_{\left\{ a+\frac{3\delta}{2}\right\} }^{\Xi_\delta}\left(z\right)\leq C\delta^\frac{1}{2}|z-a|^{-\frac{1}{2}},
\end{equation}
\begin{equation}
\label{eq:beurling-Lambda}
\mathrm{hm}_{A}^{\Xi_\delta}\left(a+\tfrac{3\delta}{2}\right)\leq C{\delta^\frac{1}{2}}(\mathrm{dist}(a;A))^{-\frac{1}{2}},
\end{equation}
where $\mathrm{hm}^{\Xi_\delta}_A(z)$ denotes the discrete harmonic measure of the set $A$ in $\Xi_\delta$ viewed from $z$, i.e. the probability for the simple random walk started at $z$ to reach $A$ before it hits the boundary of $\Xi_\delta$.
\end{lem}
\begin{proof}
This easily follows from \cite{lawler-limic} and simple reversibility arguments for random walks.
\end{proof}
Below we also need some additional estimates for the discrete harmonic measure $\mathrm{hm}_{\left\{ a+\frac{3\delta}{2}\right\} }^{\Xi_\delta}$ and its ``discrete derivatives''.
\begin{lem}
\label{lem: hm-estimate-La+grad}
(i) If $z\in\Xi_\delta$ is such that $\Re\mathfrak{e}\,(z-a)<0$ and $|\Im\mathfrak{m}\,(z-a)|\le\frac{1}{2}|\Re\mathfrak{e}\,(z-a)|$ (i.e., $z$ is close to the cut $L_a$), then
\begin{equation}
\label{eq: hm-near-La}
\mathrm{hm}_{\left\{ a+\frac{3\delta}{2}\right\} }^{\Xi_\delta}\left(z\right)\le
C\delta^\frac{1}{2}|\Im\mathfrak{m}\,(z-a)|\cdot|z-a|^{-\frac{3}{2}}\,.
\end{equation}
(ii) For all edges $(zz')\in \Xi_\delta$, one has
\begin{equation}
\label{eq: hm-gradients}
\left|\,\delta^{-1}\cdot(\mathrm{hm}_{\left\{ a+\frac{3\delta}{2}\right\} }^{\Xi_\delta}\left(z'\right) - \mathrm{hm}_{\left\{ a+\frac{3\delta}{2}\right\}}^{\Xi_\delta}\left(z\right))\right|\le C\delta^\frac{1}{2}|z-a|^{-\frac{3}{2}}.
\end{equation}
(iii) After the normalization by the value $\vartheta(\delta)$ at the proper lattice approximation of the point $a+1$, the function $\vartheta(\delta)^{-1}\mathrm{hm}_{\left\{ a+\frac{3\delta}{2}\right\}}$ converges to $\Re\mathfrak{e}\,[1/\sqrt{z-a}]$ uniformly on compact subsets of $\C\setminus L_a$.  Moreover, this convergence holds true in $C^1$-sense meaning that the ``discrete derivatives'' (\ref{eq: hm-gradients}) converge to the corresponding partial derivatives as well.
\end{lem}
\begin{proof}
(i) To prove (\ref{eq: hm-near-La}), note that the probability for the random walk started at $z$ to leave the ball of radius $\frac{1}{2}|z-a|$ around $z$ before hitting $\partial\Xi_\delta$ is $O(|\Im\mathfrak{m}\,(z-a)|\cdot|z-a|^{-1})$, and if that happened the probability to hit $a+\frac{3\delta}{2}$ is still uniformly bounded by $O(\delta^\frac{1}{2}|z-a|^{-\frac{1}{2}})$ due to the Beurling estimate (\ref{eq: hm-beurling}).

\smallskip

\noindent (ii) The estimate (\ref{eq: hm-gradients}) for the discrete derivatives follows from (\ref{eq: hm-beurling}), (\ref{eq: hm-near-La}) and the (discrete) Harnack estimate (e.g., see Proposition~2.7 in \cite{chelkak-smirnov-i}), applied to the ball of radius $\frac{1}{2}|z-a|$ (or $\frac{1}{2}|\Im\mathfrak{m}\,(z-a)|$, if $z$ is close to $L_a$) around~$z$.

\smallskip

\noindent (iii) This is essentially a special case of Theorem~3.13 in~\cite{chelkak-smirnov-i} which claims the convergence of discrete Poisson kernels normalized at some inner point to their continuous counterparts. The fact that our domain is unbounded plays no role here since
the positive discrete harmonic function $\vartheta(\delta)^{-1}\mathrm{hm}_{\left\{ a+\frac{3\delta}{2}\right\}}$ is uniformly bounded in any annulus $z:r\le|z-a|\le R$, $0<r<R$. Indeed, if it is big at some point $v$ in the annulus, then, by the maximum principle, it is also big along some nearest-neighbor path running from $v$ to $a+\frac{3\delta}{2}$. The discrete harmonic measure of such a path as seen from $a+1$ is bounded from below by a constant, leading to a contradiction, since our function equals $1$ at this point by definition of $\vartheta(\delta)$. In particular, $\vartheta(\delta)^{-1}\mathrm{hm}_{\left\{ a+\frac{3\delta}{2}\right\}}$ is uniformly bounded near any circle around $a$, and so everywhere outside this circle too, thus preventing appearance of the term $\Re\mathfrak{e}\,{\sqrt{z-a}}$ in the limit (which corresponds to the mass at $\infty$ and plays a crucial role in the proof of Lemma~\ref{def_r_delta} given below).
\end{proof}

\subsubsection{Construction of the full-plane spinor $F_{[\C_\delta,a]}$}
We are now ready to construct $F_{[\C_\delta,a]}$, as announced in Lemma~\ref{lem: def_f_C}.

\begin{proof}[\textbf{Proof of Lemma \ref{lem: def_f_C}}]
Let $\Xi_{\pm}$ denote two copies of $\C_\delta\backslash L_a$. We first define the real component $F^1_{[\C_\delta,a]}$ on $\mathcal{V}^{1}_{\Xi_{\pm}}$ as
\begin{equation}
\label{F-1-as-hm}
F^1_{[\C_\delta,a]}(z)= \pm\mathrm{hm}^{\Xi_\pm}_{\left\{ a+\frac{3\delta}{2}\right\} }\left(z\right),\quad z\in\mathcal{V}_{\Xi_\pm}^1.
\end{equation}
Since $F^1_{[\C_\delta,a]}$ vanishes identically on $L_a$, by identifying the upper side of this cut in $\Xi_+$ with the lower side in $\Xi_-$ and vice versa, we obtain a function $F^1_{[\C_\delta,a]}$ which is defined and harmonic everywhere on $\mathcal{V}^{1}_{[\C_\delta,a]}$ except at the (two) points over $a+\frac{3\delta}{2}$. Recall that Lemma~\ref{lem: hm-estimate-La+grad}(iii) ensures the convergence
\begin{equation}
\label{F1-hm-convergence}
\frac{1}{\vartheta(\delta)}F^1_{[\C_\delta,a]}(z) ~\underset{\delta\to0}{\longrightarrow}~ \Re\mathfrak{e}\,\frac{1}{\sqrt{z-a}},
\end{equation}
 which holds true uniformly on compact subsets of $\C\setminus L_a$, as well as the convergence of ``discrete derivatives'' of $F^1_{[\C_\delta,a]}(z)$ to the  corresponding partial derivatives of $\Re\mathfrak{e}\,[{1}/{\sqrt{z-a}}]$.

We then define the imaginary component $F^i_{[\C_\delta,a]}:\mathcal{V}^{i}_{[\C_\delta,a^\rightarrow]}=\mathcal{V}^{i}_{[\C_\delta,a]}\backslash \{a+\frac{\delta}{2}\}\to i\R$ as discrete harmonic conjugate of $F^1_{[\C_\delta,a]}$, that is, by integrating the identity (\ref{eq: disc_holom}) along paths on $\mathcal{V}^{i}_{[\C_\delta,a^\rightarrow]}$ starting from, say, one of the two fibers of the point $a+\frac{5\delta}{2}$. Since $F^1_{[\C_\delta,a]}$ is harmonic, its discrete harmonic conjugate is well-defined at least on the universal cover of $[\C_\delta,a^\rightarrow]$. Further, let $\varpi$ denote some simple loop in $\mathcal{V}^{i}_{\C_\delta}$, starting and ending at $a+\frac{5\delta}{2}$, symmetric with respect to the horizontal line $\{a+x,x\in\R\}$, and surrounding the singularity (so, it lifts to $[\C_\delta,a^\rightarrow]$ as a path connecting two different fibers of $a+\frac{5\delta}{2}$). It follows from the (anti)symmetry of $F^1_{[\C_\delta,a]}$ with respect to the cut $L_a$ (note that $\varpi$ changes the sheet once passing across $L_a$), that
the total increment of $F^i_{[\C_\delta,a]}$ along $\varpi$ is zero. Thus, $F^i_{[\C_\delta,a]}$ vanishes at both fibers of the point $a+\frac{5\delta}{2}$, and hence inherits the spinor property of its discrete harmonic conjugate $F^1_{[\C_\delta,a]}$.

Note that $F^i_{[\C_\delta,a]}$ vanishes not only at $a+\frac{5\delta}{2}$ which was chosen above as the starting point of the discrete integration, but also everywhere on the half-line $R_a:=\{a+x,x\ge \frac{\delta}{2}\}$ since, for $b:=a+\frac{\delta}{2}+2j\delta$, $j\ge 1$, the discrete holomorphicity equation (\ref{eq: s-hol-reduced}) and the symmetry of $F^1_{[\C_\delta,a]}$ with respect to the horizontal line $\{a+x,x\in\R\}$ imply
\[
F^i_{[\C_\delta,a]}(b+2\delta)-F^i_{[\C_\delta,a]}(b)= i\cdot[F^1_{[\C_\delta,a]}(b-i\delta)-2F^1_{[\C_\delta,a]}(b)+F^1_{[\C_\delta,a]}(b+(2-i)\delta)]= \frac{i}{2}\,\Delta F^1_{[\C_\delta,a]}(b+\delta)=0.
\]
Further, the uniform estimate (\ref{eq: hm-gradients}) guarantees that $F^i_{[\C_\delta,a]}$ is uniformly bounded (just take a path of discrete integration in the definition of $F^i_{[\C_\delta,a]}$ running from $a+\frac{5\delta}{2}$ to $|z|$ and then along the  circular arc to $z$).

It is worthwhile to note that $F^i_{[\C_\delta,a]}$ also admits a discrete harmonic measure representation similar to (\ref{F-1-as-hm}). Namely, let $\Xi'_{\pm}$ denote the two sheets of $[\C_\delta,a^\rightarrow]\setminus R_a$, where signs in the notation are chosen so that $F^1_{[\C_\delta,a]}> 0$ in the upper half-plane on $\Xi'_+$ (correspondingly, in the lower half-plane on $\Xi'_-$). Then,
\begin{equation}
\label{F-i-as-hm'}
F^i_{[\C_\delta,a]}(z)=\mp i\cdot\mathrm{hm}^{\Xi'_\pm}_{\{a+\frac{\delta}{2}\}}(z),\quad z\in\mathcal{V}_{\Xi'_\pm}^i.
\end{equation}
Indeed, on each of the sheets $\Xi'_\pm$ one can further extend $F^i_{[\C_\delta,a]}$ (as a harmonic conjugate of $F^1_{[\C_\delta,a]}$) to the point $a+\frac{\delta}{2}$: the only obstruction to do so on  $[\C_\delta,a]$ was that the increment of $F^i_{[\C_\delta,a]}$ along the smallest loop surrounding $a+\frac{3\delta}{2}$ would be non-zero since $F^1_{[\C_\delta,a]}$ is not harmonic at $a+\frac{3\delta}{2}$, but on $\Xi'_\pm$ this loop intersects the cut $R_a$. Therefore, the (bounded) function $F^i_{[\C_\delta,a]}$ is harmonic on $\Xi'_\pm$ and vanishes on $R_a\setminus \{a+\frac{\delta}{2}\}$, thus being proportional to $\mathrm{hm}^{\Xi'_\pm}_{\{a+\frac{\delta}{2}\}}(z)$. Finally, it follows from the symmetry arguments that
\begin{eqnarray*}
\mathrm{hm}^{\Xi'_\pm}_{\{a+\frac{\delta}{2}\}}\left(a+\left(\tfrac{3}{2}+i\right)\delta\right)-
\mathrm{hm}^{\Xi_\pm}_{\{a+\frac{\delta}{2}\}}\left(a+\tfrac{\delta}{2}\right) & = & \mathrm{hm}^{\Xi_\pm}_{\{a+\frac{3\delta}{2}\}}\left(a+\left(\tfrac{1}{2}+i\right)\delta\right)-
\mathrm{hm}^{\Xi_\pm}_{\{a+\frac{3\delta}{2}\}}\left(a+\tfrac{3\delta}{2}\right) \\ & = & F_{[\C_\delta,a]}^1\left(a+\left(\tfrac{1}{2}+i\right)\delta\right)-
F_{[\C_\delta,a]}^1\left(a+\tfrac{3\delta}{2}\right) \\
& = & i \cdot [F_{[\C_\delta,a]}^i\left(a+\left(\tfrac{3}{2}+i\right)\delta\right)-
F_{[\C_\delta,a]}^i\left(a+\tfrac{\delta}{2}\right)],
\end{eqnarray*}
which fixes the multiplicative constant. Again, Theorem 3.13 of \cite{chelkak-smirnov-i} implies
\begin{equation}
\label{Fi-hm-convergence}
\frac{1}{\vartheta(\delta)}F^i_{[\C_\delta,a]}(z) ~\underset{\delta\to0}{\longrightarrow}~ \Im\mathfrak{m}\,\frac{1}{\sqrt{z-a}},
\end{equation}
uniformly on compact subsets of $\C\setminus R_a$, together with the discrete derivatives. Now, following Remark \ref{rem: s-hol-extended}, we extend $F_{\C_\delta,a]}$ to $\mathcal{V}^{\lambda,\overline{\lambda}}_{[\C,a]}$ as an s-holomorphic function. The convergence (\ref{F_C-convergence}) readily follows from (\ref{F1-hm-convergence}) and (\ref{Fi-hm-convergence}) (for points lying near $L_a$, one can approximate $F^1_{[\C_\delta,a]}$ by summing discrete derivatives of $F^i_{[\C_\delta,a]}$ and vice versa near $R_a$).
\end{proof}

The following lemma explains the future role of $F_{[\C_\delta,a]}$: it has the same ``discrete singularity'' (whatever it means) as  $F_{[\Omega_\delta,a,\Ak]}$ at $a+\frac{\delta}{2}$. This allows one to handle this singularity by taking the difference $F_{[\Omega_\delta,a;\Ak]}-F_{[\C_\delta,a]}$
\begin{lem}
\label{lem: remove_sing}
The spinor
$
F_{[\Omega_\delta,a,\Ak]}^\mathrm{(a)}:=F_{[\Omega_\delta,a,\Ak]}-F_{[\C_\delta,a]},
$
extended to be zero at $a+\frac{\delta}{2}$, is s-holomorphic in any simply-connected neighborhood of $a$ which do not contain other branching points $a_1,\dots, a_k$.
\end{lem}
\begin{proof}
By Lemma \ref{lem: sing_f_Omega}, it suffices to show that
$
 \mathsf{P}_{i\mathbb{R}}\left[F_{[\C_{\delta},a]}\left(a+\frac{1\pm i}{2}\delta\right)\right]  =  \mp i
$ on the sheet $\Xi_+$. The considerations given in the previous proof show that $F_{[\C_\delta,a]}$ can be extended in an s-holomorphic way at $a+\frac{\delta}{2}$ such that
\[
 \mathsf{P}_{i\mathbb{R}}\left[F_{[\C_{\delta},a]}\left(a+\tfrac{1\pm i}{2}\delta\right)\right]  =  - i\cdot \mathrm{hm}^{\Xi'_+}_{\{a+\frac{\delta}{2}\}}(a+\tfrac{\delta}{2})=-i,\quad \text{for}~~a+\tfrac{1\pm i}{2}\delta\in \Xi'_+.
\]
However, by definition, $\Xi'_+$ coincides with $\Xi_+$ in the upper half-plane, but with $\Xi_-$ in the lower half-plane.
\end{proof}

\subsubsection{Construction of the harmonic spinor $G_{[\C_\delta,a]}$}
\label{sub:G-construction} Now we ``integrate'' the real component of $F_{[\C_\delta,a]}$ in order to construct a discrete counterpart of the harmonic spinor $\Re\mathfrak{e}\,\sqrt{z-a}$, as announced in Lemma~\ref{def_r_delta}. Along the way, we also prove the double-sided estimate (\ref{eq: vartheta-estimate}) of the normalizing factor $\vartheta(\delta)$. Note that the upper bound is given by discrete Beurling estimate (\ref{eq: hm-beurling}), so we need to prove the lower bound only.

\begin{proof}[\textbf{Proof of Lemma \ref{def_r_delta} and the estimate (2.16).}]
We use notation introduced above. Define, for $z\in\mathcal{V}^{1}_{\C_\delta}$,
\begin{equation}
G_{[\C_{\delta},a]}(z):=
\delta\cdot{\sum\limits_{j=0}^{\infty}} F^1_{[\C_\delta,a]}(z-2j\delta).
\label{eq: def_r}
\end{equation}
(due to the estimate (\ref{eq: hm-near-La}), this sum always converges). Clearly, $G_{[\C_{\delta},a]}$ vanishes on the left cut $L_a$. We are going to prove that it is harmonic \emph{everywhere} inside $\Xi_+$ (including the point $a+\frac{3\delta}{2}$).
For $z$ outside $R_a$, the harmonicity of $G_{[\C_{\delta},a]}$ readily follows from the harmonicity of $F^1_{[\C_{\delta},a]}$, so let $z\in R_a$.
Denote by $\mathcal{R}_N(z):=\mathcal{V}^{1}_{\C_\delta}\cap\{z+w:-2N\delta\leq\Re\mathfrak{e}\, w,\, \Im\mathfrak{m}\, w \leq 2N\delta\}$ the (sufficiently large) square centered at $z$ and write the discrete Green formula:
\begin{equation}
\label{x-estim}
\sum\limits_{j=0}^{N} \Delta F^1_{[\C_\delta,a]}(z-2j\delta)=
\sum\limits_{w\in \mathcal{R}_N(z)} \Delta F^1_{[\C_\delta,a]}(w)=
\sum\limits_{(ww')\in \partial\mathcal{R}_N(z)} (F^1_{[\C_\delta,a]}(w')-F^1_{[\C_\delta,a]}(w)).
\end{equation}
Let $N$ be large enough so that $|w-a|\ge N\delta$ for all boundary edges $(ww')$ of $\mathcal{R}_N(z)$. Then, it immediately follows from the estimate (\ref{eq: hm-gradients}) that the last sum is $O(N\cdot \delta^{\frac{3}{2}}(N\delta)^{-\frac{3}{2}})=O(N^{-\frac{1}{2}})$. Passing to a limit as $N\to\infty$, we conclude that
\[
\Delta G_{[\C_\delta,a]}(z)=\delta\sum\limits_{j=0}^{\infty} \Delta F^1_{[\C_\delta,a]}(z-2j\delta)=0,
\]
so $G_{[\C_\delta,a]}$ is indeed discrete harmonic in $\Xi_+$. Moreover, since it vanishes on the cut $L_a$ one can harmonically extend $G_{[\C_\delta,a]}$ to the whole double cover $\mathcal{V}^1_{[\C_\delta,a]}$ by symmetry.


Let $\nu(\delta)$ denote the value of $G_{[\C_\delta,a]}$ in a lattice approximation of the point $a+1$. Arguing as in the proof of Lemma~\ref{lem: hm-estimate-La+grad}(iii), we see that
\begin{equation}
\label{G-convergence}
\frac{1}{\nu(\delta)}G_{[\C_\delta,a]}(z) ~\underset{\delta\to0}{\longrightarrow}~ \Re\mathfrak{e}\,{\sqrt{z-a}}
\end{equation}
(since $G_{[\C_\delta,a]}$ vanishes everywhere on $L_a$ and remains bounded near $0$ by the maximum principle, in this case the limiting positive harmonic function should be proportional to $\Re\mathfrak{e}\,{\sqrt{z-a}}$, and the multiplicative normalization is fixed at $a+1$). Moreover, this convergence is uniform on compact subsets of $\C\setminus L_a$ and the same convergence holds true for discrete derivatives. In particular, this implies
\[
\frac{1}{\nu(\delta)}\cdot\frac{1}{2}F^1_{[\C_\delta,a]}(z) ~\underset{\delta\to0}{\longrightarrow}~ \partial_x\Re\mathfrak{e}\,{\sqrt{z-a}}=\frac{1}{2}\Re\mathfrak{e}\,\left[\frac{1}{\sqrt{z-a}}\right],
\]
thus yielding $\nu(\delta)\sim\vartheta(\delta)$ as $\delta\to 0$. Note that this allows us to give a simple proof of the lower bound in (\ref{eq: vartheta-estimate}): since discrete harmonic functions $(\nu\left(\delta\right))^{-1}G_{[\C_\delta,a]}$ are uniformly bounded near the unit circle around $a$ and vanish identically on $L_a$, discrete Beurling estimate(\ref{eq:beurling-Lambda}) implies
\[
\frac{\delta}{\nu\left(\delta\right)}=\frac{G_{[\C_\delta,a]}\left(a+\frac{3\delta}{2}\right)}{\nu\left(\delta\right)}\le C\sqrt\delta
\]
(and $\nu(\delta)\sim\vartheta(\delta)$ as $\delta\to 0$). Finally, the convergence (\ref{G-convergence}) near $L_a$ follows from the convergence of $F^1_{[\C_\delta,a]}(z)$, since the tails in definition (\ref{eq: def_r}) are uniformly small due to (\ref{eq: hm-near-La}).
\end{proof}


\subsection{\label{sub: riemann-problem}The boundary value problem for spinors}
In this section we reformulate the Riemann-type boundary value problem for holomorphic spinors (both discrete and continuous) using primitives of their squares. Note that this approach is not completely straightforward, since the square of a holomorphic function, in general, does not have discrete primitive. However, it was noted in~\cite{smirnov-i} that one can naturally define the real part of the primitive, using the \emph{s-holomorphicity} of observables. Moreover, a technique developed in \cite{chelkak-smirnov-ii} (see, in particular, sections 3.4 and 3.5 therein) allows one to treat this real part essentially if it was a harmonic function. Below, we summarize the tools we will use. We warn the reader that all our definitions are equivalent to those of \cite{chelkak-smirnov-ii} \emph{after the multiplication of the spinor by $\sqrt{i}$}, which means that imaginary part, sub-/super-harmonicity and positivity of functions and their (inner) normal derivatives used in \cite{chelkak-smirnov-ii} should be replaced by real part, super-/sub-harmonicity and negativity, respectively.

\subsubsection{\label{sub:discrete-integration}Discrete integration of the squared spinor observables} Let $\Delta_{\delta}^{\circ}$ be the standard (unnormalized) discrete Laplacian
acting on functions $H_{\delta}^\circ:\mathcal{V}_{\Omega_{\delta}}^{\circ}\to\mathbb{R}$ (which are defined on faces of $\Omega_\delta$):
\[
\Delta_{\delta}^{\circ}H_{\delta}^\circ\left(z\right)~:=~\sum_{w\sim z}\left(H_{\delta}^\circ\left(w\right)-H_{\delta}^\circ\left(z\right)\right),\quad z\in\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\circ},
\]
where the sum is over the four neighbors $w\in{\mathcal{V}}_{\Omega_{\delta}}^{\circ}$
of $z$. Similarly, for functions $H_{\delta}^\bullet:{\mathcal{V}}_{\Omega_{\delta}}^{\bullet}\to\mathbb{R}$ defined on vertices of $\Omega_\delta$, let $\Delta_{\delta}^{\bullet}$ denote the slightly modified discrete Laplacian:
\[
\Delta_{\delta}^{\bullet}H_{\delta}^\bullet\left(z\right) ~:=~ \sum_{w\sim z}c\left(w,z\right)\left(H_{\delta}^\bullet\left(w\right)-H_{\delta}^\bullet\left(z\right)\right),\quad z\in\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\bullet},
\]
where the conductance $c\left(w,z\right)$ is equal to $1$ for inner edges (i.e., for $w\in\mathrm{Int}\mathcal{V}_{\Omega_{\delta}}^{\bullet}$) and is equal to ${2}({\sqrt{2}-1})$ for the boundary edges (see Section~3.6 in \cite{chelkak-smirnov-ii} or \cite{duminil-copin-hongler-nolin} for the reason of this ``boundary modification'' of $\Delta_{\delta}^{\bullet}$).
\begin{prop}
\label{prop:int-square-spinor} For an s-holomorphic spinor observable $F_\delta=F_{\left[\Omega_\delta,a;\Ak\right]}:[\Omega_\delta,a^\rightarrow;a_1,\dots,a_k]\to\C$ satisfying boundary conditions (\ref{eq: discrete_bc}), one can define a real-valued function $H_\delta=H_{\left[\Omega_{\delta},a;\Ak\right]}:{\mathcal{V}}_{\Omega_{\delta}}^{\bullet\circ}\to\mathbb{R}$ which is a discrete analogue of the primitive $\Re\mathfrak{e}\int(F_\delta(z))^{2}dz$, so that the following properties are fulfilled:
\begin{itemize}
\item for any adjacent $w\in{\mathcal{V}}_{\Omega_{\delta}}^{\circ}$ and $v\in\mathcal{V}_{\Omega_{\delta}}^{\bullet}$, one has
\begin{equation}
\label{eq:H-def}
H_\delta^\circ\left(w\right)-H_\delta^\bullet\left(v\right)  =  2\delta\left|F_\delta\left(\tfrac{1}{2}(w+v)\right)\right|^{2},
\end{equation}
where $\tfrac{1}{2}(w+v)\in\mathcal{V}_{\Omega_{\delta}}^{\mathrm{c}}$ is the corner between $v$ and $w$ (according to Lemma~\ref{lem: sing_f_Omega}, we set $\left|F_\delta\left(a+\frac{\delta}{2}\right)\right|:=1$);
\item $H_\delta$ satisfies Dirichlet boundary conditions:
$H_\delta^\circ\left(v\right)=0$ for any $w\in\partial\mathcal{V}_{\Omega_{\delta}}^{\circ}$, and
$H_\delta^\bullet\left(v\right)=0$ for any $v\in\partial\mathcal{V}_{\Omega_{\delta}}^{\bullet}$;
\item $H_\delta^\bullet$ has a ``negative inner normal derivative'', i.e. $H_\delta\left(v\right)\leq 0$ for any vertex $v\in\mathcal{V}_{\Omega_{\delta}}^{\bullet}$ adjacent to a boundary vertex;
\item $H_\delta^{\circ}$ is $\Delta_{\delta}^{\circ}$-subharmonic
on $\mathcal{V}_{\Omega_{\delta}}^{\circ}\setminus\left\{ a,\Ak\right\}$, while
$H_\delta^\bullet$ is $\Delta_{\delta}^{\bullet}$-superharmonic on $\mathcal{V}_{\Omega_{\delta}}^{\bullet}\setminus\left\{a+\delta\right\}$.
\end{itemize}
\end{prop}
\begin{proof}
All the claims follow directly from the results of Section~3.3 in \cite{chelkak-smirnov-ii}. Since all listed properties are local, the spinor nature of $F_\delta$ plays no role here (note that the right-hand side of (\ref{eq:H-def}) does not depend on the sheet).
\end{proof}

\begin{rem}
\label{rem:maximum}
By construction, $H^{\circ}_\delta(w)\geq H^{\bullet}_\delta(v)$ for adjacent $w$ and $v$. Combined with super- and sub-harmonicity, this implies a maximum principle for $H_\delta$: if $\Omega'_\delta\subset \Omega_\delta$ does not contain $a$ (respectively, any of $a,\Ak$), then \[
\min\limits_{\Omega'_\delta}H_\delta=\min\limits_{\partial\Omega'_\delta}H_\delta^{\bullet}\quad \text{(respectively,}\; \max\limits_{\Omega'_\delta}H_\delta=\max\limits_{\partial\Omega'_\delta}H_\delta^\circ).
\]
Moreover, if $\mathrm{hm}_A(z)$ denotes the discrete harmonic measure of a set $A$ in $\Omega'_\delta$ viewed from $z$, then
\begin{eqnarray*}
H_\delta(z)\geq (1-\mathrm{hm}^\bullet_A(z))\min\limits_{\partial\Omega'_\delta}H_\delta^{\bullet}+ \mathrm{hm}^\bullet_A(z)\min\limits_{A}H_\delta^{\bullet};\\
H_\delta(z)\leq (1-\mathrm{hm}^\circ_A(z))\max\limits_{\partial\Omega'_\delta}H_\delta^{\circ}+\mathrm{hm}^\circ_A(z)\max\limits_{A}H_\delta^{\circ}.
\end{eqnarray*}
\end{rem}
\begin{rem}
\label{rem:fix_singularity}
The subharmonicity of $H^{\circ}_\delta$ fails at $a,a_1,\dots,a_k$ because $F_{[\Omega_\delta,a;a_1\dots a_k]}$ branches at those points, while the superharmonicity of $H^{\bullet}_\delta$ fails at $a+\delta$ because of the discrete singularity of $F_{[\Omega_\delta,a;a_1\dots a_k]}$ which is not defined at $a+\frac{\delta}{2}$. In particular, Lemma \ref{lem: remove_sing} implies that the function
\[
\textstyle H_{\delta}^{\mathrm{(a)}}=H_{[\Omega_\delta,a;a_1\dots,a_k]}^{\mathrm{(a)}}:= \Re\mathfrak{e}\int(F^\mathrm{a}_\delta(z))^2dz,\quad F_\delta^\mathrm{(a)}=F_{[\Omega_\delta,a;a_1\dots,a_k]}-F_{[\C_\delta,a]},
\]
(defined in the same way as $H_{\delta}$ accordingly to Proposition~\ref{prop:int-square-spinor}) is superharmonic on $\mathcal{V}^\bullet_{\Omega_\delta}$ and subharmonic on $\mathcal{V}^\circ_{\Omega_\delta}\setminus\{a\}$ everywhere in a neighborhood of $a$. Moreover, since $F_\delta^\mathrm{(a)}(a+\frac{\delta}{2})=0$ on both sheets, the values of $F_\delta^\mathrm{(a)}$ at the nearby corners $a\pm\frac{\delta}{2},a\pm \frac{i\delta}{2}$ and midedges $a\pm\frac{1\pm i}{2}\delta$ satisfy the \mbox{s-holomorphicity} conditions as if it were nonbranching at $a$. Thus, $H_{\delta}^{\mathrm{(a)}}$ is subharmonic at the branching point $a\in\mathcal{V}^\circ_{\Omega_\delta}$ too.
\end{rem}

\subsubsection{Integration of squared spinors in the continuous setup.}
We now give a characterization of the continuous spinors solving boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}) in terms of the primitives of their squares. In the next section, this characterization will be used in the proofs of main convergence results.

\begin{prop}
\label{prop: obs_uniqueness}
 Let $\Omega$ be a simply-connected domain, and suppose a holomorphic spinor $f=f_{[\Omega,a;a_1,\dots,a_k]}$ solves the boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}) (or is defined according to Remark \ref{rem: spinor_rough}(ii), if $\Omega$ is not smooth). Define two harmonic functions
 \[
 \textstyle h:=\Re\mathfrak{e}\int(f(z))^2dz\quad\text{and}\quad h^{\mathrm{(a)}}:=\Re\mathfrak{e}\int(f(z)-f_{\C,a}(z))^2dz,
 \]
where $f_{\C,a}(z):=1/\sqrt{z-a}$. Then, the following holds true:
\begin{enumerate}
 \item $h$ is a single-valued function in $\Omega\backslash\{a,\Ak\}$, continuous up to $\partial\Omega$, which satisfies Dirichlet boundary conditions $h\equiv \text{const}$ on $\partial \Omega$ (since $h$ is defined up to an additive constant, we assume that $h\equiv 0$ on $\partial \Omega$);
 \item there is no point $z_0$ on $\partial\Omega$ such that $h(z)\geq 0$ in a neighborhood of $z_0$;
 \item $h$ is bounded from below in a neighborhood of each $\Ak$;
 \item $h^{\mathrm{(a)}}$ is single-valued and bounded in a neighborhood of $a$.
\end{enumerate}
Moreover, if $h$ and $h^\mathrm{(a)}$ satisfy (1)--(4), then $f$ solves the boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}).
\end{prop}
\begin{proof}
Note that, being integrated, the covariance property (\ref{eq: obs_covariance}) claims the conformal invariance of both $h$ and $h^\mathrm{(a)}$. As the  properties (1)--(4) are preserved under conformal mappings too, we will further assume that $\partial \Omega$ is smooth. Note that the property (\ref{def:spinor-bdry}) is equivalent to (1) and (2): it states that $f^2(z)\nu_\mathrm{out}(z)$ is positive on the boundary, which yields $\partial_{\tau_\mathrm{out}(z)}h \equiv 0$ (where $\tau(z)$ denotes a tangent vector) and $\partial_{\nu_\mathrm{out}(z)}h \geq 0$ everywhere on $\partial\Omega$. A straightforward integration of the asymptotics of $f$ near $a_j$ and $a$ given by the conditions (\ref{def:spinor-1}) and (\ref{def:spinor-2}), respectively, gives
\begin{equation}
\begin{array}{rcll}
h(z)& = &- c_j\log|z-a_j|+O(1), & z\rightarrow a_j,\\
h^\mathrm{(a)}(z)& = &\log|z-a|+O(1), & z\rightarrow a,
\end{array}
\end{equation}
for some $c_j\ge 0$, giving (3) and (4). Vice versa, (3),(4) imply (\ref{def:spinor-1}),(\ref{def:spinor-2}) by differentiating and taking the square root.
\end{proof}

\subsection{Convergence of discrete observables away from singularities} \label{sub:convergence-away-sing} In this section we prove the convergence of (properly normalized) discrete spinor observables to their continuous counterparts on compact subsets of $\Omega\setminus\{a,\Ak\}$.

\begin{proof}[\textbf{Proof of Theorem \ref{thm:obs-away}}]
Let the function
\[
H_{\delta}:=\Re\mathfrak{e}\int\left(\frac{1}{\vartheta\left(\delta\right)}F_\delta(z)\right)^{2}dz,\quad F_\delta=F_{\left[\Omega_\delta,a;\Ak\right]},
\]
be defined on $\mathcal{V}_{\Omega_{\delta}}^{\bullet\circ}$ accordingly to Proposition \ref{prop:int-square-spinor}. Given $\epsilon>0$, define
\[
\Omega_\delta(\epsilon):=\Omega_\delta\cap \{z:\mathrm{dist}(z;\{a,\Ak\})>\epsilon\}.
\]

 Assume for a moment that, for any $\epsilon>0$, the functions $H_{\delta}$ remain uniformly bounded on $\Omega_\delta(\epsilon)$ by some constant $C(\epsilon)$ as $\delta\to 0$. Then by Theorem 3.12 in \cite{chelkak-smirnov-ii}, the functions $(\vartheta(\delta))^{-1}F_\delta$ are equicontinuous on $\Omega_\delta(\epsilon)$. Thus, by passing to a subsequence and applying the diagonal process, we can assume that $(\vartheta\left(\delta\right))^{-1}F_\delta$ tends to a limit $\tilde{f}$ and $H_{\delta}\rightarrow \tilde{h}:=\Re\mathfrak{e}\int \tilde{f}^2$ uniformly on compact subsets of $\Omega\backslash\{a,\Ak\}$. Our goal is to check that $\tilde{f}$ satisfies the properties (1) -- (4) given in Proposition \ref{prop: obs_uniqueness}, then the  uniqueness proven in Remark~\ref{rem: spinor_rough}(i) implies $\tilde{f}=f_{[\Omega,a;\Ak]}$.

Clearly, $\tilde{f}$ is a holomorphic spinor on $[\Omega,a,\Ak]$. By superhamonicity of $H^\bullet_\delta$, we have
\[
\min\limits_{|z-a_j|<2\epsilon}H_\delta(z)=\min\limits_{\epsilon<|z-a_j|<2\epsilon}H^\bullet_\delta(z),
\]
 thus $\tilde{h}$ is bounded from below in the neighborhoods of $\Ak$, so the property (3) holds true. By maximum principle for $H_{\delta}$ (see Remark \ref{rem:maximum}), taking into account that $H_\delta\equiv0$ on $\partial\Omega_\delta$, we have that
\begin{equation}
\label{eq: F_boundary}
\begin{array}{rcl}
 H^{\bullet}_{\delta}(z) & \geq & -C(\epsilon)(1-\mathrm{hm}^{\Omega_\delta(\varepsilon)}_{\partial \Omega_\delta}(z))\\[3pt]
 H^{\circ}_{\delta}(z) & \leq & C(\epsilon)(1-\mathrm{hm}^{\Omega_\delta(\varepsilon)}_{\partial \Omega_\delta}(z))
 \end{array}
\end{equation}
Since $\mathrm{hm}^{\Omega_\delta(\varepsilon)}_{\partial \Omega_\delta}(z)\longrightarrow 1$ uniformly in $\delta$ as $z$ approaches the boundary of $\Omega_\delta$, this implies $\tilde{h}\equiv 0$ on $\partial\Omega$, giving (1). Moreover, thanks to Remark~6.3 in~\cite{chelkak-smirnov-ii}, we also have (2): there is no point on $\partial \Omega$  such that $\tilde{h}\geq 0$ in its neighborhood.

Consider now the following function, discussed (up to multiplicative normalization) in Remark~\ref{rem:fix_singularity}:
\[
H_\delta^\mathrm{(a)}:=\Re\mathfrak{e}\int \left(\frac{1}{\vartheta\left(\delta\right)}F_\delta^\mathrm{(a)}(z)\right)^2dz,\quad F_\delta^\mathrm{(a)}=F_\delta-F_{[\C_\delta,a]},
\]
which is well-defined in the disc $\{z:|z-a|<r\}$ provided that $r$ is small enough. Since both $(\vartheta(\delta))^{-1}F_\delta$ and $(\vartheta(\delta))^{-1}F_{[\C_\delta,a]}$ uniformly converge to $\tilde{f}$ and $f_{[\C,a]}$ (see Lemma~\ref{lem: def_f_C}) on compact subsets of $A_r:=\{z:0<|z-a|<r\}$, $H_\delta^\mathrm{(a)}$ converges to $\tilde{h}^\mathrm{(a)}:=\Re\mathfrak{e}\int^z(\tilde{f}(w)-f_{[\C,a]}(w))^2dw$ everywhere in $A_r$. By Remark~\ref{rem:fix_singularity} (sub- and super-harmonicity of $H_\delta^\mathrm{(a)}$ on $\mathcal{V}^\circ$ and $\mathcal{V}^\bullet$ near the point $a$), the functions $H_\delta^\mathrm{(a)}$ are uniformly bounded in $A_r$, so $\tilde{h}^\mathrm{(a)}$ is bounded in $A_r$ too, which concludes the proof of the property (4). Therefore, $\tilde{f}=f_{\left[\Omega,a;\Ak\right]}$.

It remains to justify the uniform boundedness of $H_{\delta}$ away from $a,\Ak$. On the contrary, suppose that
\[
M_\delta(\epsilon):=\max\limits_{\Omega_\delta(\epsilon)}|H_{\delta}|\underset{\delta\rightarrow 0}{\longrightarrow}\infty
\]
for some $\epsilon>0$ and along some subsequence of $\delta$'s. Then the (re-)normalized functions $(M_\delta(\epsilon))^{-1}H_{\delta}$ are uniformly bounded in $\Omega_\delta(\varepsilon)$, and thus $(M_\delta(\epsilon))^{-\frac{1}{2}}(\vartheta(\delta))^{-1}F_\delta$ and $(M_\delta(\epsilon))^{-1}H_{\delta}$ have subsequential limits $\tilde{f}$ and $\tilde{h}=\Re\mathfrak{e}\int\tilde{f}^2$, holomorphic and harmonic in $\Omega_\delta(\epsilon)$, respectively. An important observation, proven in Lemma \ref{non-vanish} below, is that $\tilde{h}$ cannot be identically zero. This implies that for any $0<\epsilon_1<\epsilon$ we have $M_\delta(\epsilon_1)\le CM_\delta(\epsilon)$ with some constant $C$ independent of $\delta$.

Hence, applying the diagonal procedure, we may assume that $(M_\delta(\epsilon))^{-\frac{1}{2}}(\vartheta(\delta))^{-1}F_\delta$ tends to a limit $\tilde{f}$ (and $(M_\delta(\epsilon))^{-1}H_{\delta}$ tends to $\tilde{h}=\Re\mathfrak{e}\int \tilde{f}^2$) uniformly on \emph{each} $\Omega_\delta(\epsilon_1)$, $0<\epsilon_1<\epsilon$. Arguing as above, we see that $\tilde{h}$ is harmonic in $\Omega\backslash \{a,\Ak\}$, satisfies Dirichlet boundary conditions, has positive outer normal derivative, and is bounded from below near $\Ak$. Moreover, the last step of the proof given above shows that
  \[
  \tilde{h}^\mathrm{(a)}=\lim\limits_{\delta\rightarrow 0} (M_\delta(\epsilon))^{-1}(\vartheta(\delta))^{-2}\Re\mathfrak{e}\int (F_\delta(z)-F_{[\C_{\delta},a]}(z))^2dz= \lim\limits_{\delta\rightarrow 0} (M_\delta(\epsilon))^{-1}(\vartheta(\delta))^{-2}\Re\mathfrak{e}\int (F_\delta(z))^2dz=\tilde{h}
  \]
is also bounded in a neighborhood of $a$ (one can neglect $F_{[\C_{\delta},a]}(z)$ in the last expression since $(\vartheta(\delta))^{-1}F_{[\C_{\delta},a]}(z)$ tends to $f_{[\C,a]}$ and $M_\delta(\epsilon)\to\infty$). By the maximum principle, this yields $\tilde{h}\equiv 0$, which is a contradiction.
\end{proof}

\begin{lem}
 \label{non-vanish}
In the notation of the proof above, none of the subsequential limits of $(M_\delta(\epsilon))^{-1}H_{\delta}$ is identically zero.
\end{lem}
\begin{proof}
 Suppose by contradiction that $(M_\delta(\epsilon))^{-1}H^{\delta}\rightarrow 0$ uniformly on compact subsets of $\Omega_\delta(\epsilon)$. Let $z^\mathrm{max}_\delta$ be the point of $\Omega_\delta(\epsilon)$ where the maximum $M_\delta(\epsilon)$ of $|H_\delta|$ is attained. By sub- and super-harmonicity of $H^{\circ}$ and $H^{\bullet}$ respectively, $z_\delta^\mathrm{max}$ belongs to one of the discrete ``circles'' $\ell_a(\epsilon):=\{z:\epsilon\leq|z-a|\leq\epsilon+5\delta\}$, $\ell_{a_j}:=\{z:\epsilon\leq|z-a_j|\leq\epsilon+5\delta\}$ (recall that $H_\delta$ vanishes on the boundary). By passing to a subsequence, we can assume that $z_\delta^\mathrm{max}\rightarrow z^\mathrm{max}$.

Suppose that $z^\mathrm{max}\in \ell_{a_j}(\epsilon)$ for some $1\leq j\leq k$. Since $H^\bullet$ is superharmonic inside $\ell_{a_j}(2\epsilon)$ and tends to zero uniformly on $\ell_{a_j}(2\epsilon)$ by our assumption, its minimum $m_\delta^j(2\epsilon)$ inside $\ell_{a_j}(2\epsilon)$ also tends to zero. Thus, $z^\mathrm{max}_\delta\in \mathcal{V}^{\circ}_{\Omega_\delta}$ and $H^\circ_\delta(z^\mathrm{max})=M_\delta(\epsilon)$. Hence, by subharmonicity of $H^\circ_\delta$, we can find a discrete nearest-neighbor path $\gamma^\circ:= \{z^\mathrm{max}_\delta=z_1\sim z_2\sim \dots\}\subset\mathcal{V}_{\Omega_\delta}^\circ$ with $M_\delta(\epsilon)\le\dots\le H^\circ_\delta(z_{i})\le H^\circ_\delta(z_{i+1})\le\dots$, which may only end up at $a_j$, where the subharmonicity fails. By Remark~3.10 in~\cite{chelkak-smirnov-ii}, the functions $H^\bullet_\delta-m_\delta^j(2\epsilon)$ and $H^\circ_\delta-m_\delta^j(2\epsilon)$ are uniformly comparable at adjacent points inside $\ell_{a_j}(\epsilon)$. Since $m_\delta^j(2\epsilon)\to 0$ as $\delta\to 0$, while $M_\delta(\epsilon)\to \infty$, we have $H^\bullet_{\delta}(z)\ge cM_\delta(\epsilon)$ for some absolute constant $c>0$ and all $z\in\gamma^\bullet$, where $\gamma^\bullet$ is the set of vertices adjacent to $\gamma^\circ$. Due to the maximum principle (see Remark~\ref{rem:maximum}), we infer that, for any $z$,
$$
H^\bullet_\delta(z)\geq cM_\delta(\epsilon)\mathrm{hm}_\gamma^\bullet(z)+(1-\mathrm{hm}_\gamma^\bullet(z))m_\delta^j(2\epsilon),
$$
where $\mathrm{hm}_\gamma^\bullet(z)$ is the discrete harmonic measure of $\gamma$ viewed form $z$ in $\{z:|z-a|\leq 2\epsilon\}$. But if $z$ is close enough (independently of $\delta$) to $z^\mathrm{max}$, then, by discrete Beurling estimate, $\mathrm{hm}_\delta(z)$ is close to one. This shows that, for such $z$, $(M_\delta(\epsilon))^{-1}H^\bullet_\delta(z)\ge c'>0$ for an absolute constant $c'$, and hence $(M_\delta(\epsilon))^{-1}H_\delta(z)$ cannot tend to zero in $\Omega(\epsilon)$.

It remains to treat the case $z^\mathrm{max}\in \ell_a(\epsilon)$. Consider the function
\[
\textstyle (M_\delta(\epsilon))^{-1}H_\delta^\mathrm{(a)}= (M_\delta(\epsilon))^{-1}(\vartheta(\delta))^{-2}\Re\mathfrak{e}\int(F_\delta(z)-F_{[\C_{\delta},a]}(z))^2dz.
\]
Note that it tends to zero on $\ell_{a_j}(2\epsilon)$, since one can neglect the term $F_{[\C_{\delta},a]}(z)$ (recall that
$(\vartheta(\delta))^{-1}F_{[\C_{\delta},a]}(z)\to f_{[\C,a]}$ and $M_\delta(\epsilon)\rightarrow \infty$). Therefore, by Remark~\ref{rem:fix_singularity} and the maximum principle, it also tends to zero on $\ell_{a_j}(\epsilon)$, so we consequently derive that each of the functions
\[
\frac{F_\delta- F_{[\C_{\delta},a]}}{(M_\delta(\epsilon))^{1/2}\vartheta(\delta)}, \quad  \frac{F_\delta}{M_\delta(\epsilon))^{1/2}\vartheta(\delta)}\quad\text{and}\quad \frac{H_\delta}{M_\delta(\epsilon)}
\]
tends to zero uniformly on $\ell_{a_j}(\epsilon)$. In particular, $1=(M_\delta(\epsilon))^{-1}|H_\delta(z^\mathrm{max}_\delta)|\rightarrow 0$, which is a contradiction.
\end{proof}


\subsection{Analysis near the singularities}\label{sub:convergence-near-sing}
We now pass to the most delicate part of our analysis: matching
the second-order terms in the values $F_\delta(a+\frac{\delta}{2})$ with the second coefficient in the expansion of the continuous spinor near the ``source'' $a$.

\begin{proof}[\textbf{Proof of Theorem \ref{thm:localization-near-a}}]
Let $\mathcal{R}_{a}$ denote the reflection with respect to the axis $\left\{ a+x,x\in\mathbb{R}\right\}$ and $\Lambda_\delta$ be a small neighborhood of $a$ in $\Omega_{\delta}\cap\mathcal{R}_{a}\left(\Omega_{\delta}\right)$. Denote by $\Lambda_\delta^+$ the sheet of a double cover $[\Lambda_\delta,a^\rightarrow]\setminus L_a$ such that $F_\delta(a+\frac{3\delta}{2})>0$ on this sheet, where $F_\delta=F_{[\Omega_\delta,a;\Ak]}$ and $L_a=\{a+x,x\le \frac{\delta}{2}\}$ as before. We define a \emph{real-valued} function $S_\delta:\mathcal{V}^1_{\Lambda_\delta^+}\to\R$ as
\[
S_{\delta}:=\frac{1}{\vartheta\left(\delta\right)}\left(\frac{F_\delta+F^{(\mathcal{R})}_\delta}{2}- F_{[\C_\delta,a]}-2\Re\mathfrak{e}\,\mathcal{A}\cdot G_{[\C_\delta,a]}\right),
\]
where $F_\delta^{(\mathcal{R})}=F_{[\mathcal{R}_{a}(\Omega_\delta),a,\mathcal{R}_{a}(a_1),\dots, \mathcal{R}_{a}(a_k)]}$, $\mathcal{A}=\mathcal{A}_{\left[\Omega,a,\Ak\right]}$ and the full-plane functions $F_{[\C_\delta,a]}$, $G_{[\C_\delta,a]}$ were constructed in Section~\ref{sub:full-plane}. Recall that $F_{[\C_\delta,a]}\left(a+\frac{3\delta}{2}\right)=1$,  $G_{[\C_\delta,a]}\left(a+\frac{3\delta}{2}\right)=\delta$ and $F_\delta^{(\mathcal{R})}(a+\frac{3\delta}{2})=F_\delta(a+\frac{3\delta}{2})$ by symmetry. Thus,
\begin{equation}
\label{x-near-sing-1}
S_{\delta}(a+\tfrac{3\delta}{2})=(\vartheta\left(\delta\right))^{-1}\left(F_\delta(a+\tfrac{3\delta}{2})-1-2\Re\mathfrak{e}\,\mathcal{A}\cdot \delta\right)
\end{equation}
and our goal is to estimate this value. Note that $S_\delta$ vanishes on the cut $L_a$: both $F_{[\C_\delta,a]}$ and $G_{[\C_\delta,a]}$ vanish by construction, and $F^{(\mathcal{R})}_\delta=-F_\delta$ on $L_a$ due to the spinor property ($F_\delta$ changes the sign between opposite sides of $L_a$, since they belong to different sheets). Note that $S_\delta$ is discrete harmonic in $\mathcal{V}^1_{\Lambda_\delta^+}\setminus\{a+\frac{3\delta}{2}\}$ since all terms are discrete harmonic, and discrete harmonic at $a+\frac{3\delta}{2}$ also, due to Lemma~\ref{lem: remove_sing}. Therefore, for any (small) $r>0$, discrete Beurling estimate (\ref{eq: hm-beurling}) implies
\begin{equation}
\label{x-near-sing-2}
S_{\delta}(a+\tfrac{3\delta}{2})\le
C{\delta^\frac{1}{2}}\cdot r^{-\frac{1}{2}}\max_{\ell_a(r)}|S_\delta|,
\end{equation}
where $\ell_a(r)$ denotes the ``discrete circle'' $\{z:r\le|z-a|\le r+5\delta\}$. It follows from Theorem~\ref{thm:obs-away} and Lemmas~\ref{lem: def_f_C},~\ref{def_r_delta} that, for any $r>0$, one has
\[
S_\delta  ~\underset{\delta\to0}{\longrightarrow}~
s:=\Re\mathfrak{e}\,[\tfrac{1}{2}(f+f^{(\mathcal{R})})- f_{[\C,a]}]-2\Re\mathfrak{e}\,\mathcal{A}\cdot g_{[\C,a]},
\]
uniformly near the circle $\{z:|z-a|=r\}$, where $f=f_{[\Omega,a;\Ak]}$, $f^{(\mathcal{R})}=f_{[\mathcal{R}_a(\Omega),a;\mathcal{R}_a(a_1),\dots,\mathcal{R}_a(a_k)}$, $f_{[\C,a]}(z)= 1/\sqrt{z-a}$ and $g_{[\C,a]}(z)= \Re\mathfrak{e}\,\sqrt{z-a}$. Recall that, by definition of the coefficient $\mathcal{A}$, one has
\[
f-f_{[\C,a]}=2\mathcal{A}\sqrt{z-a}+O(|z-a|^\frac{3}{2}),\quad z\to a.
\]
It is easy to check that $f^{(\mathcal{R})}(z)\equiv \overline{f(\mathcal{R}_a(z))}$ (since this spinor solves the corresponding boundary value problem), hence
\[
f^{(\mathcal{R})}-f_{[\C,a]}=2\overline{\mathcal{A}}\sqrt{z-a}+O(|z-a|^\frac{3}{2}),\quad z\to a.
\]
Thus, we arrive at $s(z)=O(|z-a|^\frac{3}{2})$ as $z\to a$, which means
\begin{equation}
\label{x-near-sing-3}
r^{-\frac{1}{2}}\max_{\ell_a(r)}|S_\delta| ~\underset{\delta\to0}{\longrightarrow}~ r^{-\frac{1}{2}}\max_{\ell_a(r)}|s|=O(r).
\end{equation}
Combining (\ref{x-near-sing-1})--(\ref{x-near-sing-3}), one concludes that, for any given $r>0$,
\[
F_\delta(a+\tfrac{3\delta}{2})-1-2\Re\mathfrak{e}\,\mathcal{A}\cdot \delta \le C\vartheta(\delta)\delta^\frac{1}{2}r,
\]
if $\delta$ is small enough. Since $\vartheta(\delta)=O(\delta^{\frac{1}{2}})$ by (\ref{eq: vartheta-estimate}), and $r$ can be chosen arbitrary small, this yields (\ref{eq:localization-near-a}). All estimates are uniform with respect to $a,\Ak$ at definite distance from the boundary and each other.
\end{proof}

To prove Theorem~\ref{thm:localization-near-b}, we need to analyze the discrete spinor $F_{\left[\Omega_\delta,a;b\right]}$ near the point $b$.
In contrast to Theorem \ref{thm:localization-near-a}, where second-order information near $a$ was extracted, here we only need to match
the first-order coefficients. Note that the situation is slightly different from the first-order analysis near $a$, as $F_{\left[\Omega_\delta,a;b\right]}$
does not have an explicit discrete singularity, remaining s-holomorphic near the branching $b$, while its limit blows up at $b$. Still, the strategy for the identification resembles the one used above and appeals to a symmetrization with respect to the horizontal line passing through $b$.

\begin{proof}[\textbf{Proof of Theorem~\ref{thm:localization-near-b}}]
Let $\mathcal{R}_{b}$ denote the reflection with respect to the axis $\left\{b+x:x\in\mathbb{R}\right\}$ and $\Lambda_\delta\subset\Omega_{\delta}\cap\mathcal{R}_{b}\left(\Omega_{\delta}\right)$ be a small neighborhood of $b$. Fix the sheet $\Lambda_\delta^+=\Lambda_\delta\setminus L_b$ on a double cover $[\Omega_\delta,a;b]$ so that $F_\delta(b+\frac{\delta}{2})=i\mathcal{B}_\delta$ with $\mathcal{B}_\delta>0$ (recall that the values of discrete spinors on $\mathcal{V}^i_{\C_\delta}$ are purely imaginary), and do the same for the continuous spinor: fix a sheet so that $\Im\mathfrak{m}\,f_{[\Omega,a;b]}(b+x)>0$ as $x\downarrow 0$ (see (\ref{eq: expansion-b})).

Let $W_\delta:=\mathrm{hm}_{\{b+\frac{\delta}{2}\}}(\,\cdot\,):\C^i_\delta\setminus L_b\to\R$ denote the harmonic measure of $b+\frac{\delta}{2}$ in the slitted discrete plane $\C_\delta^{i}\setminus L_b$ and
\[
T_{\delta}:=\frac{1}{\vartheta\left(\delta\right)}\left(\frac{F_\delta+F^{(\mathcal{R})}_\delta}{2}- i\mathcal{B}_\delta\cdot W_\delta\right)~:\mathcal{V}^i_{\Lambda_\delta^+}\to i\R,
\]
where $F_\delta=F_{[\Omega_\delta,a;b]}$ and $F^{(\mathcal{R})}_\delta=F_{[\mathcal{R}_b(\Omega),\mathcal{R}_b(a);b]}$. By symmetry, $F^{(\mathcal{R})}_\delta(b+\frac{\delta}{2})=i\mathcal{B}_\delta$ (if one fixes the sheet for $F_\delta^{(\mathcal{R})}$ by the same condition $F^{(\mathcal{R})}_\delta(b+\frac{\delta}{2})\in i\R_+$). Note that the function $W_\delta$ is harmonic in $\Lambda_\delta^+$ and vanishes everywhere on the cut $L_b$, including the point $b+\frac{\delta}{2}$: for $z\in L_b\setminus\{b+\frac{\delta}{2}\}$, one has  $F_\delta+F^{(\mathcal{R})}_\delta=0=W_\delta$ due to symmetry reasons and the spinor property of $F_\delta$, and $T_\delta(b+\frac{\delta}{2})=0$ since $W_\delta(b+\frac{\delta}{2})=1$.

We have to prove that $\mathcal{B}_\delta\to\mathcal{B}=\mathcal{B}_{[\Omega,a;b]}$ as $\delta\to 0$. By passing to a subsequence, we may assume that $\mathcal{B}_\delta\to\widetilde{\mathcal{B}}$ for some $\widetilde{\mathcal{B}}\in[0,+\infty]$. If $\widetilde{\mathcal{B}}$ is finite, then  Theorem~\ref{thm:obs-away} and Lemma~\ref{lem:discrete-beurling}(iii) give
\begin{equation}
\label{x-near-sing-b1}
T_\delta(z) ~\underset{\delta\to0}{\longrightarrow}~ t(z):=\Im\mathfrak{m}\,\left[\frac{f(z)+f^{(\mathcal{R})}(z)}{2}- i\widetilde{B}\cdot\Re\mathfrak{e}\,\frac{1}{\sqrt{z-b}}\right],
\end{equation}
uniformly on compact subsets of $\{z:0<|z-b|\le r\}$, where $f=f_{[\Omega,a;b]}$ and $f^{(\mathcal{R})}(z)\equiv f_{[\mathcal{R}_b(\Omega),\mathcal{R}_b(a);b]}(z)\equiv -\overline{f(\mathcal{R}_b(z))}$ on the sheet $\Lambda_\delta^+$. Note that
\begin{equation}
\label{x-near-sing-b2}
t(z)=i(\mathcal{B}-\widetilde{\mathcal{B}})\cdot\Re\mathfrak{e}\,\frac{1}{\sqrt{z-b}}+O(|z-b|^\frac{1}{2}),\quad z\to b.
\end{equation}
Therefore, $\widetilde{\mathcal{B}}\ne\mathcal{B}$ contradicts to the maximum principle for the discrete harmonic function $T_\delta$: (\ref{x-near-sing-b1}) and (\ref{x-near-sing-b2}) imply that, for sufficiently small $\delta$, its values inside of the circle $\ell_b(r)$ around $b$ are bigger then on $\ell_b(r)$.

The similar argument works, if $\widetilde{\mathcal{B}}=+\infty$: in this case one would have
\[
\mathcal{B}_\delta^{-1}T_\delta(z)~\underset{\delta\to0}{\longrightarrow}~ -i\cdot\Re\mathfrak{e}\,\frac{1}{\sqrt{z-b}},
\]
which contradicts to the maximum principle for $T_\delta$, just as before.
\end{proof}


\renewcommand{\thesection}{A}
\section{Appendix. Spinors in the half-plane and decorrelations identities} \label{sec:appendix-continuous-computations}

In this Appendix we explicitly construct the continuous spinor which satisfies Definition~\ref{def:spinor-bounded} in the half-plane. Recall that, along with (\ref{def:spinor-bdry})--(\ref{def:spinor-2}), one should also impose the condition $f_{[\bH,a;\Ak]}(z)=O(|z|^{-1})$ at infinity, so that after a conformal mapping to a bounded domain with smooth boundary the spinor remains bounded. We argue that the construction of $f_{[\bH,a;a_1,\dots,a_k]}$ boils down to a system of $k+1$ linear equations, which is always non-degenerate. Consequently, we derive the formulae for $\mathcal{A}_{[\bH,a;\Ak]}$ and, by integration, the continuous correlation functions. Although for small $k$ it is possible to check by ugly brute force computations that the functions we get coincide with those predicted by the Conformal Field Theory arguments (recall that the computations of $1$-point and $2$-point functions were done in Section~\ref{sub:12pts-computations}), we prove this coincidence only for some particular configurations when all $a,\Ak$ lie on the imaginary axis. This turns out to be sufficient to prove Lemma \ref{lem: norm_cont_corr}.


\subsection{\label{sub:loc-exp-cts-spinors}The explicit formulae for spinors}


Let us introduce some notation. For $w\in\bH$, denote
\[
 p_{w}(z):=(z-w)(z-\overline{w}).
\]
Given $a,\Ak\in\bH$, we look for $f_{[\bH,a;a_1,\dots,a_k]}$ in the following form:
\[
f_P(z):=e^{\frac{\pi i}{4}}\cdot \frac{P(z)}{\sqrt{p_{a}(z)p_{a_1}(z)\dots p_{a_k}(z)}},
\]
where $P(z)=\sum_{n=0}^k q_nz^n$ is a polynomial of degree $k$ with \emph{real} coefficients. Clearly, $f_P$ is a holomorphic spinor on $[\bH,a;\Ak]$ which satisfies the boundary condition (\ref{def:spinor-bdry}), hence we have to determine the coefficients $q_0,q_1,\dots,q_k$ from the conditions (\ref{def:spinor-1}) and (\ref{def:spinor-2}). Since $e^{\frac{\pi i}{4}}/\sqrt{p_{a_m}(z)}=1/\sqrt{(2\Im\mathfrak{m}\, a_m)(z-a_m)}+O(|z-a_m|^\frac{1}{2})$ as $z\to a_m$, these conditions can be rewritten as
\begin{equation}
\label{Appendix-system}
\begin{array}{rl}
\sum\limits_{n=0}^{k}M_{0,n}(a;\Ak)q_n=1,&\\
\sum\limits_{n=0}^{k}M_{m,n}(a;\Ak)q_n=0,&\quad m=1,\dots,k,
\end{array}
\end{equation}
where
\begin{equation*}
M_{m,n}=  M_{m,n}(a;\Ak) ~:=~ \Re\mathfrak{e}\,\left[a_m^n\cdot \left[{\textstyle\prod_{l\ne m}p_{a_l}(a_m)}\right]^{-\frac{1}{2}}\right],\quad 0\le m,n\le k
\end{equation*}
(here and below we use the notation $a_0=a$). If this linear system is non-degenerate, the Cramer's rule implies
\begin{equation}
\label{Appendix-qn}
q_n=q_n(a;\Ak)=(-1)^n\det_{m\ne 0,s\ne n}\left[M_{m,s}\right]\cdot D^{-1},\quad \text{where}\quad D={\det_{0\le m,s\le k}\left[M_{m,s}\right]}.
\end{equation}
\begin{prop} For any $k+1$ different points $a,\Ak\in\bH$, the linear system (\ref{Appendix-system}) has a unique solution which is given by (\ref{Appendix-qn}). The spinor
\begin{equation}
\label{Appendix-f,P}
f_{[\C,a;\Ak]}(z):=e^\frac{\pi i}{4}\cdot \frac{P_{a;\Ak}(z)}{\sqrt{p_{a}(z)p_{a_1}(z)\dots p_{a_k}(z)}},\qquad P_{a;\Ak}(z):=\sum_{n=0}^kq_nz^n,
\end{equation}
solves the boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}) and satisfies the additional regularity condition $f_{[\C,a;\Ak]}(z)=O(|z|^{-1})$ at infinity. Moreover, all the spinors $f_{[\C,a;\Ak]}(z)$ are uniformly bounded, if $z$ is $\epsilon$-away from $a,\Ak$, and
\begin{equation}
\mathcal{A}_{\bH}(a;\Ak)=-\frac{1}{4(a-\overline{a})}-\frac{1}{4}\sum\limits_{s=1}^{k}\left[\frac{1}{a-a_s}+\frac{1}{a-{\overline{a}_s}}\right]+ \left.\frac{\partial_zP_{a;\Ak}(z)}{2P_{a;\Ak}(z)}\right|_{z=a}.
\label{log_deriv_explicit}
\end{equation}
\label{prop: spinor_explicit}
\end{prop}
\begin{proof}
On the contrary, let the system (\ref{Appendix-system}) be degenerate. Then, a nontrivial solution to its homogeneous counterpart gives rise to a spinor $f_P$ that satisfies (\ref{def:spinor-bdry}) and (\ref{def:spinor-1}), as well as $\Re\mathfrak{e}\, [\lim_{z\rightarrow a}\sqrt{z-a}f_P(z)]=0$, and thus vanishes identically, since
\begin{equation}
\label{Appendix-f^2-int}
0\le i^{-1}\int_{\R}(f_P(z))^2dz=2\pi\lim\limits_{z\rightarrow a}(z-a)(f_P(z))^2+2\pi \sum_{s=1}^k \lim\limits_{z\rightarrow a_s}(z-a_s)(f_P(z))^2\le 0,
\end{equation}
which is a contradiction. Thus, (\ref{Appendix-system}) is non-degenerate and has a solution (\ref{Appendix-qn}). Let $f_{[\C,a;\Ak]}$ be given by (\ref{Appendix-f,P}), thus satisfying the boundary conditions (\ref{def:spinor-bdry}), and
\begin{equation}
\label{Appendix-beta}
\textstyle \beta_s=\beta_s(a;\Ak):=\lim_{z\rightarrow a}\sqrt{z-a}f_{[\C,a;\Ak]}(z),\quad 0\le s\le k.
\end{equation}
Note that the linear system (\ref{Appendix-system}) means $\Re\mathfrak{e}\,\beta_0=1$ and $\Re\mathfrak{e}\,\beta_s=0$ for all other $s$, hence the conditions (\ref{def:spinor-2}) are satisfied. The contour integration argument (\ref{Appendix-f^2-int}) ensures that
$\beta_0^2-(|\beta_1|^2+\dots +|\beta_k|^2)\ge 0$. In particular, $\beta_0$ should be purely real (and so $\beta_0=1$), i.e. the condition (\ref{def:spinor-2}) is satisfied too. Moreover, one has
\[
h_{[\C,a;\Ak]}(z):=\Re\mathfrak{e}\,\int(f_{[\C,a;\Ak]}(z))^2dz = G_\bH(z;a)-\sum_{s=1}^k \beta_s^2G_\bH(z;a_s),
\]
where $G_\bH(z;a)$ is the Green's function in the half-plane. Thus, $(f_{[\C,a;\Ak]}(z))^2=\frac{1}{2}[\partial_z G_\bH(z;a)-\sum_{s=1}^k\beta_s^2 \partial_zG_\bH(z;a_s)]$ are uniformly bounded, since the gradients of the Green's functions are uniformly bounded, if $z$ is $\epsilon$-away from the singularities. Finally, the formula (\ref{log_deriv_explicit}) readily follows from (\ref{Appendix-f,P}).
\end{proof}

\subsection{\label{sub:corr-explicit}Conformal Field Theory predictions and the particular case $a,\Ak\in i\R_+$}

It was predicted by the methods of Conformal Field Theory \cite{burkhard-guim} that in the continuum limit, the spin correlation functions take the following form:
\begin{equation}
\label{eq:cont_cft}
 \langle \sigma_{a_0} \dots \sigma_{a_k}\rangle_{\bH,\text{CFT}}^{+}=\prod\limits_{m=0}^{k}\frac{1}{\left(2\Im\mathfrak{m}\, a_m\right)^{\frac18}}\left[2^{-\frac{k+1}{2}}\sum\limits_{\mu_0=\pm 1,\dots,\mu_k=\pm 1} \, \prod\limits_{s<m}\left|\frac{a_s-a_m}{a_s-\overline{a}_m}\right|^{\frac{\mu_s\mu_m}{2}}\right]^{\frac12}.
\end{equation}
\begin{rem}
\label{rem: cft-predictions}
Recall that we have considered two simplest cases $k=0$ and $k=1$ in Section~\ref{sub:12pts-computations}. Namely, we found the explicit solutions (\ref{eq: 1pt-spinor}) and (\ref{eq: 2pts-spinor}) to the boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}), computed their asymptotics near $a$ and obtained the corresponding coefficients $\mathcal{A}_{\bH}(a)$ and $\mathcal{A}_\bH(a;a_1)$. Then, one can directly check that those coefficients coincide with the logarithmic derivatives of the quantities (\ref{eq:cont_cft}), thus establishing this Conformal Field Theory prediction. Unfortunately, this procedure becomes much harder as $k$ grows. Having in hand explicit formulae given in the previous Section, one \emph{can} compute $\mathcal{A}_\bH(a;\Ak)$ and integrate the answer, but it is not easy to \emph{check} whether it coincides with (\ref{eq:cont_cft}) or not. Note that the case $k=2$ is still doable by hand. In this case, it is easy to check that, for a proper real constant $C$, the spinor
\[
f_{[\bH,a_0;a_1,a_2]}(z):= Ce^\frac{\pi i}{4}\cdot\frac{c_{02}p_{a_1}(z)+c_{01}p_{a_{2}}(z)-c_{12}p_{a_0}(z)}{\sqrt{p_{a_0}(z)p_{a_1}(z)p_{a_2}(z)}},\quad c_{sm}:=|a_s-a_m||a_s-\overline{a}_m|,
\]
satisfies the conditions (\ref{def:spinor-bdry})--(\ref{def:spinor-2}), which allows us to verify the CFT prediction (\ref{eq:cont_cft}) for the 3-point function by brute force computations. In the next lemma we prove (\ref{eq:cont_cft}) for all $k$ in the special case when all point are purely imaginary, which is sufficient for our purposes.
\end{rem}

\begin{lem} If $a_m=iw_m\in i\R_+$ for all $m=0,\dots,k$, then
 $\partial_{w_0}\log\langle\sigma_{a_0} \dots \sigma_{a_k}\rangle_{\bH,\mathrm{CFT}}^{+}= i\mathcal{A}_\bH(a_0;a_1\dots a_k)$.
\label{lem: CFT_eq_ours}
\end{lem}
\begin{proof}
We can rewrite the function $\langle \sigma_{a_0} \dots \sigma_{a_k}\rangle_{\bH,\text{CFT}}^{+}$ given by (\ref{eq:cont_cft}) in the following form:
\[
\frac{2^{-\frac{3}{8}(k+1)}}{\prod_{m=0}^{k}{\left(\Im\mathfrak{m}\, a_m\right)^{\frac18}} \prod_{s<m}{\left|a_s-a_m\right|^{\frac14}\left|a_s-\overline{a}_m\right|^{\frac{1}{4}}}}
\,\biggl[\,\sum\limits_{\mu_0=\pm 1,\dots,\mu_k=\pm 1}\, \prod\limits_{s<m}|a_s-a_m|^{\frac{1+\mu_s\mu_m}{2}}|a_s-\overline{a}_{m}|^{\frac{1-\mu_s\mu_m}{2}}\biggr]^{\frac12}
\]
Observe that the logarithmic derivative of the first factor exactly matches the first two terms of (\ref{log_deriv_explicit}). Hence it suffices to show that
\begin{eqnarray}
\label{eq: imline_reduced}
\partial_{w_0} \log\biggl[\,\sum\limits_{\mu_0=\pm 1,\dots,\mu_k=\pm 1}\prod\limits_{s<m}|w_s-w_m|^{\frac{1+\mu_s\mu_m}{2}}|w_s+w_m|^{\frac{1-\mu_s\mu_m}{2}}\biggr
=& \left.i\partial_{z} \log P_{a_0;\Ak}(z)\right|_{z=a_0} \notag \\
=& \left.\partial_{w} \log P_{a_0;\Ak}(iw)\right|_{w=w_0}.
\end{eqnarray}
Let, say, $w_0<w_1<\dots <w_k$, the other cases are treated by similar computations. Then,
\begin{eqnarray}
\label{Appendix-x1}
\sum\limits_{\mu_0=\pm 1,\dots,\mu_k=\pm 1}\;\prod\limits_{s<m}|w_s-w_m|^{\frac{1+\mu_s\mu_m}{2}}|w_s+w_m|^{\frac{1-\mu_s\mu_m}{2}} & = & \sum\limits_{\mu_0=\pm 1,\dots,\mu_k=\pm 1}\;\prod\limits_{s<m} (w_m-\mu_s\mu_m w_s) \notag \\
= \sum\limits_{\mu_0=\pm 1,\dots,\mu_k=\pm 1}\;{\prod_{m=0}^k\mu_m^m}\prod\limits_{s<m} (\mu_m w_m-\mu_sw_s) & = & \sum\limits_{\mu_0=\pm 1,\dots,\mu_k=\pm 1} \;{\prod_{m=0}^k\mu_m^m\det\limits_{0\le m,n\le k}\,[(\mu_m w_m)^n]} \notag \\
\sum\limits_{\mu_0=\pm 1,\dots,\mu_k=\pm 1} \;\det\limits_{0\le m,n\le k}\,[\mu_m^{m+n}w_m^n] & = & \det\limits_{0\le m,n\le k}\,[(1+(-1)^{m+n})w_m^n].
\end{eqnarray}
On the other hand, $p_{a_l}(a_m)=\sign(l\!-\!m)\cdot|w_l-w_m||w_l+w_m|$, thus
\[
\quad M_{m,s}= \Re\mathfrak{e}\,\left[(iw_m)^s\cdot i^{m}R_m\right],\qquad \text{where}\quad R_m=\left[{\textstyle\prod_{l\ne m}} |w_l-w_m||w_l+w_m|\right]^{-\frac{1}{2}}
\]
are real and do not depend on $n$. Note that $\Re\mathfrak{e}\, [i^{m+s}]=\frac{1}{2}(1+(-1)^{m+s})i^{m+s}$.
Therefore, (\ref{Appendix-qn}) reads
\begin{eqnarray*}
q_n=D^{-1}\cdot(-1)^n\det_{m\ne 0,s\ne n}\left[M_{m,s}\right] & = & C\cdot (-1)^n \det_{m\ne 0,s\ne n}\left[i^{s+m}(1+(-1)^{m+s})w_m^s\right]\\
& = & C\cdot (-1)^ni^{k(k+1)-n} \det_{m\ne 0,s\ne n}\left[(1+(-1)^{m+s})w_m^s\right]\\
& = & (-1)^\frac{k(k+1)}{2}C\cdot (-i)^n \det_{m\ne 0,s\ne n}\left[(1+(-1)^{m+s})w_m^s\right],
\end{eqnarray*}
where $C=D^{-1}2^{-k}R_1\dots R_m$, and
\begin{equation}
\label{Appendix-x2}
P_{a_0;\Ak}(iw)= \pm C\cdot\sum_{n=0}^k~w^n\det_{m\ne 0,s\ne n}\left[(1+(-1)^{m+s})w_m^s\right] = \pm\tfrac{1}{2}C \cdot \det_{0\le m,n\le k}\left[(1+(-1)^{n+m})w_m^n\right],
\end{equation}
since $\det_{m\ne 0,s\ne n}\left[(1+(-1)^{n+m})x_{m,n}\right]=0$ for odd $n$ and any $x_{m,n}$. Thus, (\ref{eq: imline_reduced}) readily follows from (\ref{Appendix-x1}) and (\ref{Appendix-x2}).
\end{proof}


\subsection{Decorrelation identities and proof of Lemma \ref{lem: norm_cont_corr}}\label{sub:decorrelations-continuous}

We start by proving that the half-plane spinors $f_{\bH,a;\Ak}$ behave in a continuous fashion, when one of the points $\Ak$ approaches the real line and ``disappears'' there.

\begin{lem}
\label{lem: f_tends}
As $a_k\to\partial\bH$, one has $f_{[\bH,a;a_1,\dots,a_k]}(z)\to f_{[\bH,a;a_1,\dots,a_{k-1}]}(z)$ and $\mathcal{A}_{[\Omega,a;a_1,\dots,a_k]}\to \mathcal{A}_{[\Omega,a;a_1,\dots,a_{k-1}]}$,
uniformly with respect to the positions of points $a,a_1,\dots,a_{k-1},z$, provided they are at least $\epsilon$-away from the boundary and each other.
\end{lem}
\begin{proof}
This can be easily proven by compactness arguments, but we prefer to give an explicit construction which also allows one to estimate the convergence rate, if necessarily. Denote
\[
g:=f_{[\bH,a;a_1,\dots,a_{k-1}]}(z)\cdot r_{a_k}(z),\quad \text{where}\quad r_{a_k}(z):=\frac{z-\Re\mathfrak{e}\,a_k}{\sqrt{(z-a_k)(z-\overline{a}_k)}}
\]
By definition, $\widetilde{f}$ satisfies the spinor property on $[\C,a;\Ak]$, the boundary conditions (\ref{def:spinor-bdry}) and $g(z)=O(|z|^{-1})$ at infinity, but the conditions (\ref{def:spinor-1}), (\ref{def:spinor-2}) fail since the values $r_{iy}(a_s)$ are nonreal. Let $\beta_s^{(k-1)}=\beta_s(a;a_1,\dots,a_{k-1})$ be defined by (\ref{Appendix-beta}), and
\[
\begin{array}{rl}
\beta_0 :=& \lim_{z\to a}\sqrt{z-a}\,g(z)= \beta_0^{(k-1)}\cdot r_{a_k}(a)=r_{a_k}(a),\\
\beta_s :=& \lim_{z\to a_s}\sqrt{z-a_s}\,g(z)= \beta_s^{(k-1)}\cdot r_{a_k}(a_s) \\
\beta_k :=& \lim_{z\to a_k}\sqrt{z-a_k}\,g(z) = f_{[\bH,a;a_1,\dots,a_{k-1}]}(a_k)\cdot (\frac{1}{2}\Im\mathfrak{m}\,a_k)^\frac{1}{2},
\end{array}
\]
where $1\le s\le k-1$. Note that $g$ can be represented as
\[
g=\sum_{n=0}^k \Re\mathfrak{e}\,\beta_s\cdot f_{[\bH,a_s;a_0,\dots,\hat{a}_s,\dots,a_k]}.
\]
Indeed, the difference of two sides would solve the homogeneous version of the boundary value problem (\ref{def:spinor-bdry})--(\ref{def:spinor-2}), which is impossible since the solution of this problem is unique due to the contour integration argument (\ref{Appendix-f^2-int}). Therefore,
\[
g-f_{[\bH,a;a_1,\dots,a_k]}=(\Re\mathfrak{e}\,\beta_0\!-\!1)f_{[\bH,a;a_1,\dots,a_k]}+\sum_{s=1}^k \Re\mathfrak{e}\,\beta_s\cdot f_{[\bH,a_s;a_0,\dots,\hat{a}_s,\dots,a_k]}.
\]
Note that $\beta_0=r_{a_k}(a)\to 1$, $\Re\mathfrak{e}\,\beta_s = -\Im\mathfrak{m}\,\beta_s^{(k-1)}\Im\mathfrak{m}\,r_{a_k}(a_s)\to 0$ and $\beta_k\to 0$ as $y\to 0$, and all the spinors in the right-hand side are uniformly bounded due to Proposition~\ref{prop: spinor_explicit}. Thus, we conclude that
\[
f_{[\bH,a;a_1,\dots,a_{k}]}(z)-f_{[\bH,a;a_1,\dots,a_{k-1}]}(z)= [f_{[\bH,a;a_1,\dots,a_{k}]}(z)- g(z)] + [1-r_{a_k}(z)]f_{[\bH,a;a_1,\dots,a_{k-1}]}(z)\to 0
\]
since both terms vanish as $y\to 0$. The convergence of $\mathcal{A}_{[\Omega,a;a_1,\dots,a_k]}$ follows by the Cauchy integral formula.
\end{proof}

We are in the position to prove Lemma~\ref{lem: norm_cont_corr}.

\begin{proof}[\textbf{Proof of Lemma \ref{lem: norm_cont_corr}}]
We first define the correlation functions $\langle \sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH}^{+}$ in the half-plane. For $k=0,1$ we use the explicit formulae (\ref{1ptFct}),(\ref{2ptFcts}), and we define them to be $\exp[\int\mathcal{L}_{\Omega,k+1}]$ in the general case, where the differential form $\mathcal{L}_{\Omega,k+1}$ is given by (\ref{eq: L-def}). Note that this form is symmetric with respect to $a,a_1,\dots,a_k$ and exact on the manifold $\widetilde{\Omega}^{k+1}$ of all $(k+1)$-tuples of distinct points in $\Omega$ due to Proposition~\ref{prop: conv_ratios}. Therefore, the function $\langle \sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH}^{+}$ is symmetric with respect to $a,a_1,\dots,a_k$ and well-defined on $\widetilde{\Omega}^{k+1}$ up to a multiplicative constant which we fix so that
\begin{equation}
\label{Appendix-CFT-ours}
\langle \sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH}^{+}=\langle \sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH,\mathrm{CFT}}^{+}~~\text{for}~~a,\Ak\in i\R_+.
\end{equation}
As it was established in Lemma~\ref{lem: CFT_eq_ours}, this normalization (everywhere on the imaginary line) does not contradict to our definition of the correlation function as an exponent of the primitive of $\mathcal{L}_{\Omega,k+1}$. Therefore, (\ref{Appendix-CFT-ours}) holds true for any $a,a_1,\dots,a_k$ lying on the same vertical line: indeed, the multiplicative normalization cannot depend on the horizontal position of this line, since for such configurations the coefficients $\mathcal{A}_\bH(a_0;a_1\dots a_k)$ are purely imaginary due to Lemma~\ref{lem: CFT_eq_ours}.

Let $a\to\partial\bH$ along some vertical line, $\Ak\in\bH$ and $b_1,\dots,b_k$ be some fixed points on the same vertical line as~$a$, say, $b_s:=\Re\mathfrak{e}\,a+is$. Write
\[
\frac{\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH}^{+}} {\langle\sigma_{a}\rangle_{\bH}^{+}\langle\sigma_{a_1}\dots\sigma_{a_{k}}\rangle_{\bH}^{+}}
=\frac{\langle\sigma_a\sigma_{b_1}\dots\sigma_{b_k}\rangle_{\bH}^{+}} {\langle\sigma_{a}\rangle_{\bH}^{+}\langle\sigma_{b_1}\dots\sigma_{b_{k}}\rangle_{\bH}^{+}}\cdot
\frac{\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH}^{+}}{\langle\sigma_a\sigma_{b_1}\dots\sigma_{b_{k}}\rangle_{\bH}^{+}}\cdot
\frac{\langle\sigma_{b_1}\dots\sigma_{b_k}\rangle_{\bH}^{+}}{\langle\sigma_{a_1}\dots\sigma_{a_{k}}\rangle_{\bH}^{+}}
\]
Since $a,b_1,\dots,b_k$ are on the same vertical line, we know from the explicit formulae (\ref{eq:cont_cft}) that the first term tends to one as $\Im\mathfrak{m}\, a\to0$. Also, we have
\[
\log\left[\frac{\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH}^{+}}{\langle\sigma_a\sigma_{b_1}\dots\sigma_{b_{k}}\rangle_{\bH}^{+}}\cdot
\frac{\langle\sigma_{b_1}\dots\sigma_{b_k}\rangle_{\bH}^{+}}{\langle\sigma_{a_1}\dots\sigma_{a_{k}}\rangle_{\bH}^{+}}\right]
=
\int\limits_{(a,b_1,\dots,b_k)}^{(a,a_1,\dots,a_k)}\mathcal{L}_{\Omega,k+1}-\int\limits_{(b_1,\dots,b_k)}^{(a_1,\dots,a_k)}\mathcal{L}_{\Omega,k}.
\]
According to definition (\ref{eq: L-def}), the differential forms $\mathcal{L}_{\Omega,k+1}$ and $\mathcal{L}_{\Omega,k}$ consist of the terms
$\Re\mathfrak{e}\,\left[\mathcal{A}_\Omega(z_j;z_1,...,\hat{z}_j,...,z_k,a)dz_j\right]$ and $\Re\mathfrak{e}\,\left[\mathcal{A}_\Omega(z_j;z_1,...,\hat{z}_j,...,z_k)dz_j\right]$, respectively, which are uniformly close to each other due to Lemma~\ref{lem: f_tends}, as $(z_1,\dots,z_k)$ runs from $(a_1,\dots,a_k)$ to $(b_1,\dots,b_k)$ and $\Im\mathfrak{m}\,a\to 0$. Note that, since the convergence is uniform, it does not matter, whether $a\to\partial\bH$ along a vertical line or not.

Thus, we have defined the half-plane functions $\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\bH}^{+}$ so that decorrelation identities (\ref{eq: bdry_decor}) are fulfilled. We then define the correlation functions $\langle \sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ in a given bounded domain $\Omega$ using a conformal map $\varphi: \Omega\to \bH$ and the covariance rule (\ref{conf-cov}). The computation given in Remark \ref{rem: covariance-ratios} shows that, being defined in this way, $\log\langle \sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ is indeed a primitive of the form $\mathcal{L}_{\Omega,k+1}$. Since the covariance rule (\ref{conf-cov}) implies the conformal invariance of the ratios ${\langle\sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}}/ {\langle\sigma_{a}\rangle_{\Omega}^{+}\langle\sigma_{a_1}\dots\sigma_{a_{k}}\rangle_{\Omega}^{+}}$, one obtains decorrelation identities (\ref{eq: bdry_decor}) as $a\to\partial\Omega$ (which means $\varphi(a)\to\bH$). The second part of (\ref{eq: bdry_decor}) was proven in Remark~\ref{rem: ratio_free_plus}.

It is easy to see that conditions (\ref{eq: bdry_decor}) fix the unknown multiplicative normalization of the correlation functions uniquely: the full-plane normalization fixes the two-point function and decorellation identities inductively fix all others. Therefore, the functions $\langle \sigma_a\sigma_{a_1}\dots\sigma_{a_k}\rangle_{\Omega}^{+}$ constructed above do not depend on a particular choice of the conformal mapping $\varphi$.
\end{proof}

