\section{\label{sec:intro}Introduction}

The effect of inhomogeneity on the critical behaviour of magnetic systems has been considered in various contexts (e.g. disorder, coupling randomness, quasiperiodic structures); in particular, discrete-spin models defined on fractal topologies possess critical properties significantly different and richer than those found for translationally invariant systems \cite{gefen1,gefen2,gefen3,milan,cassi,done}. 

The interest in fractal structures is not purely theoretical: many condensed-matter systems display strong nonuniformity on all length scales and can therefore be characterized as fractal objects; examples include the backbone of percolation clusters, aggregates obtained from diffusion-limited growth processes, and absorbent surfaces.

One of the most known fractals is the Sierpinski gasket (SG), which, due to its exact decimability, allows analytical approaches; in particular, by means of renormalization group techniques, it was proved that the Ising model on the SG exhibits phase transition only at zero temperature, while at any finite temperature the system breaks into domains and loses long-range order \cite{gefen1,gefen2,gefen3,donetti}. While this result was found in the thermodynamic limit, at the mesoscopic sizes peculiar  
and interesting thermodynamic properties arise \cite{liu}. More precisely, the Ising model defined on a finite SG exhibits critical-like
 features at nonzero, low temperatures and the solution found in the thermodynamic limit turns out to be a poor approximation. This anomalous behavior has been investigated from a thermodynamic point of view and derives from long-range, slowly decaying correlations at low temperatures \cite{luscombe}.

The way such a behavior is reflected by the evolution of spin configurations is an item so far overlooked (even if, for rectangular lattices, the idea of studying cluster dynamics may be traced back to Peierls and Griffith \cite{peierls, griffith}). Given the importance of the dynamics of Ising-like clusters in many research areas, from condensed matter to biological systems \cite{appl1,appl2,appl3}, the definition and the development of proper tools for this kind of analysis would be very useful. 
Moreover, it would be particularly intriguing for inhomogeneous substrates, due to the emerging non-trivial thermodynamic behavior; on the other hand, it is just on such structures that the definition of a proper metrization or evolutionary dynamics can be more awkward. 

The interest in cluster mobility actually extends to an extremely wide class of models: indeed the Ising model on the SG may be seen as a particular realization of cellular automata on graphs, i.e. discrete time dynamical systems assigning to each node of a graph $\mathbf G$ a value chosen in a alphabet $\mathbb K $, along a rule depending only on a finite neighborhood at previous time
\cite{wolfram}. 


In this work we aim to introduce and develop proper algorithms for the quantitative characterization of cluster dynamics on graphs, and we use this approach for the Ising model on a finite SG, meant as a prototype of cellular automata on graphs. The procedure requires a projection of evolving configurations into an appropriate {\sl partition space}, where an information-based metrics (Rohlin distance) and  a method measuring the effective emergence of configurational novelty (reduction process and amplification parameter) may be naturally defined and worked out in order to focus the changing and the stable components of configurations. The  algorithmic implementations of Rohlin distance and related quantities are deeply affected by the topological features of the substrate \cite{nostro}. For instance, in previous implementations designed specifically for automata on regular lattices, the very passage from one to two dimensions yields a much higher order of complexity, and could not be exported on different substrates \cite{parti1,parti2,soc,entro}. 
On the contrary, the algorithm developed here can be directly applied to generic cellular automata for which the metric characterization of cluster dynamics gets feasible; a brief description is given in the appendix $B$.

Our investigations on the Ising SG highlight the existence of two ``critical'' temperatures, $T_I$ and $T_{II},$ demarcating three main regimes which recover, both qualitatively and quantitatively, the results of the previous thermodynamic analysis; in addition, within the above mentioned regimes, we obtain a more detailed dynamical and geometrical characterization. In particular, at very low temperatures
($T < T_I$), a long-range order is established, and the few small-sized clusters display poor overlap from one time step to the next one: this makes the distance close to zero, and the reduction ineffective. As the spot sizes start to increase, the slowness of the evolution is still such that both non-similarity and overlap between successive configurations rapidly grow.
At greater temperatures ($T_I < T < T_{II}$), large-scale correlations start to decay, clusters of all sizes appear, and overlap gets easier; then, for $T> T_{II}$, any trace of order has vanished and, even if overlaps are very important, the complexity of magnetic pattern is irreducible. The progression of such different kinds of non-similarity and dynamical overlap is well described by our parameters (distance, amplification and intersection, see below).
We will also evidence the existence of different scales controlling the disappearance of local and correlated order. 
Thus, the phenomenology provided by our method, confirms with a deeper geometric insight the peculiar critical-like behavior exhibited by the system


We finally notice that, in view of future extensions to non-equilibrium situations, we will adopt a microcanonical dynamics \cite{ACV2009}, which allows implementations even in the presence of temperature gradients.

In the next two sections, after recalling basic notations on graphs, we review some facts concerning the thermodynamic properties of the Ising model on the Sierpinski gasket (Sec.~\ref{sec:SG}), and the microcanonical dynamics working as evolutionary dynamics (Sec.~\ref{sec:Micro}). Then, we introduce general procedures for cluster identification and reduction (Sec.~\ref{sec:metrica}) and we show our results on the SG (Sec.~\ref{sec:numerics}). Finally, we present our conclusions and perspectives (Sec.~\ref{sec:con}). Technical remarks about partitions on graphs can be found in the appendices.




\section{\label{sec:SG}Ising model on the Sierpinski gasket}

A generic graph $\mathbf{G}$ is mathematically specified by the
pair $\{\Lambda, \Gamma \}$ consisting of a non-empty, countable
set of points $\Lambda$ joined pairwise by a set of links
$\Gamma$. 
The cardinality  $|\Lambda| = N$ of $\Lambda$ represents the number of sites making up the graph, i.e. its
volume. From an algebraic point of view, a graph
is completely described by its
adjacency matrix $\mathbf{A}$. Every entry of this off-diagonal, symmetric
matrix corresponds to a pair of sites, and it equals $1$ if and
only if this couple is joined by a link, otherwise it is $0$. 



Here we consider the Sierpinski gasket which can be built recursively with the following procedure: the initial state ($\mathbf{G}_0$) is a triangle and the $g$th stage $\mathbf{G}_g$ is obtained joining two of the three external corners of three $\mathbf{G}_{g-1}$ to form a bigger triangle (see Fig.~\ref{fig:sierp}). In this way, the volume of $\mathbf{G}_{g}$ is $3(3^g-1)/2$. The Gasket is obtained as the limit for $g \rightarrow \infty$ of this procedure.




\begin{figure}[tb] \begin{center}
\includegraphics[width=.30\textwidth]{sierp.eps}
\caption{\label{fig:sierp} (Color on line) Sierpinski gasket of generation
$2$ (a), $3$ (b) and $4$ (c), with volume $N=6$, $N=15$ and $N=42$,
respectively.}
\end{center}
\end{figure}

\subsection{\label{ssec:termo_SG}Thermodynamic properties of the Ising model on the SG}
The Ising model on a generic graph $\mathbf{G}=\{\Lambda, \Gamma \}$ is defined associating the spin variable $s_i = \pm 1$ to every site $i$ of the graph, and considering a nearest-neighbours interaction between points $i$ and $j$, such that $A_{ij}=1$. The Hamiltonian is therefore
\begin{equation}
H_{\mathbf{G}}(\mathbf{s},J) = - J \sum_{i=1}^N \sum_{j>i}^N A_{ij} s_i s_j,
\end{equation}
where $\mathbf{s}=\{s_i\}_{i \in \Lambda}$ denotes the magnetic configuration of the system and the coupling $J$ is assumed to be the same for any couple; in the following we will set $J \equiv 1$.

As it is well known, a magnetic model defined on a finite lattice cannot exhibit critical behaviour at nonzero temperatures; critical features can only emerge when the underlying lattice becomes infinitely large, i.e. in the thermodynamic limit. Another necessary condition in order to have a nonzero critical temperature concerns the topology of the (infinite) substrate: for Euclidean structures it was rigorously shown that the dimension must be larger than 1. Analogously, it has been shown that the discrete symmetry Ising model on finitely ramified fractals cannot have a nonzero critical temperature \cite{gefen3,luscombe,bhattacharya}.
However, the critical behaviour of one-dimensional systems and finitely ramified fractals can be markedly different, since for the latter it is further governed by additional geometric aspects such as ramification, lacunarity, and connectivity \cite{gefen1}.


The thermodynamic properties of the Ising model on the SG were studied in details in \cite{gefen1,gefen2,gefen3,milan,liu,luscombe,stosic}, where it was shown that its scale-invariant, fractal structure leads to highly cooperative correlations and, at sufficiently low temperatures, the correlation length $\xi$ becomes extremely large and slowly decaying (compared with the one-dimensional case), so that any system with size smaller than $\xi$ displays long-range order. 
Indeed, one can define an apparent magnetic transition temperature $T_1$ as the point where $\langle m^2 \rangle =0.5$; for a system of generation $g$ ($g \gg 1$),
$T_1 \approx 4J / \ln(4g)$, for example, for generation $g=5$ and $g=6$ one finds $T_1 \approx 1.335$ and $T_1 \approx 1.259$, respectively. 
Hence, as the system size is enlarged, $T_1$ diminishes slowly; more precisely, being $N=3(3^g+1)/2$ the number of nodes for a gasket of generation $g$, one has $T_1 \sim 1/ \ln[\ln(N)]$.

On the other hand, the specific heat $c$ does not display any anomaly associated with long-range order, yet it exhibits a peak at a temperature $T_2 \approx 2J$  \cite{liu}. Conversely, the ``reduced'' specific heat $\sim ~ c T^2$, (basically the derivative of the magnetic energy per link with respect to the inverse temperature \cite{luscombe}), evidences a qualitative difference between the SG and a one-dimensional system, since for the former it exhibits a peak, while for latter it grows monotonically as the temperature is increased. Indeed, for the SG, as $T$ is increased from small values, the energy decreases rather slowly as a result of a relatively large cost due to the large (bulk) coordination number; at larger temperatures it becomes progressively easier to reduce the energy and this reflects the fact that the large fluctuations begin to develop; finally, since the energy must ultimately vanish, the rate of change gets smaller and smaller \cite{luscombe}.

In conclusion, the SG displays a non-trivial thermodynamic behavior which can be summarized as follows. 
At $T< T_1$ a long-range order is established; for $T_1 < T < T_2$ a short-range order is still present with large  fluctuations on all length scales. At larger temperatures $T>T_2$ any trace of order has disappeared and a paramagnetic state is approached.




\section{\label{sec:Micro}Microcanonical Dynamics}
When studying transport properties, microcanonical dynamics are usually chosen as they allow to describe an isolated system, or its isolated bulk, without any assumption on
the equilibrium state between the system and the surrounding. 


Here we adopt a recently introduced microcanonical dynamics \cite{ACV2009}, which features a high degree of flexibility, being ergodic in any temperature range and implementable on a generic structure, even in the presence of disorder. On regular lattices (e.g. cylinder, torus) such a dynamics has already been shown to be able to lead the system to thermalized states compatible with those expected from a canonical dynamics and to allow the study of out-of-equilibrium properties \cite{ACV2009,future}.   

Although we will focus only on equilibrium regimes, the reason for choosing this dynamics is twofold: First we test its reliability on an inhomogeneous structure; second, we pave the ground for the study of transport properties on such a substrate.


In the following, we briefly resume how it works having in mind as substrate a generic graph $\mathbf{G}$ described by the adjacency matrix $\mathbf{A}$.

For each pair of connected sites, namely each link $i, j$, such that $A_{ij}=1$, besides the magnetic energy $E^m_{ij} = J s_i s_j$, we introduce a local {\it kinetic energy} $E_{ij}>0$ which is, in principle, unbounded.
Now, the dynamical rule proceeds as follows:
\begin{enumerate}
\item Start from a (discrete) distribution of energies $\{E_{ij}\}_{i,j=1,...,N}$;
\item choose randomly a link  $i \sim j$;
\item extract one over the possible four spin-configurations for the couple of sites $i,j$, and evaluate the magnetic energy variation $\Delta E^m_{ij}$ induced by the move;
\item if $\Delta E_{ij}^m \leq 0$, accept the move and increase the link energy $E_{ij}$ of $\Delta E_{ij}^m$. When $\Delta E^m_{ij} >0$, the move is accepted only if $E_{ij} \geq \Delta E_{ij}^m$ and the link energy is consequently decreased of $\Delta E_{ij}^m$; otherwise  the move is not accepted and the link energy is not updated.
\end{enumerate}

It is worth remarking that $\Delta E_{ij}^m$ allows for energy variations occurred on the link  $i \sim j$ as well as on those pertaining to links adjacent to sites $i$ or $j$:
$$
\Delta E_{ij}^m = 2 s_i \delta_i  \sum_{k \in \Lambda} A_{ik} s_k + 2 s_j \delta_j  \sum_{k \in \Lambda} A_{jk} s_k - 4 s_i s_j \delta_i \delta_j,
$$
where $\delta_i = 1$ if the $i$th spin has undergone a spin flip, otherwise it is zero.
It is therefore clear that, due to the discreteness of the system and to the fact that $J$ is constant over all links, both $E^m_{ij}$ and $E_{ij}$ are discrete variables: the former can only assume two different values corresponding to the aligned and non aligned configurations of the adjacent spins $i$ and $j$; the latter can only assume integer values deriving by proper combinations of the pertaining $\Delta E_{ij}^m$.

We also notice that $E_{ij}$ works as an additional degree of freedom and the above dynamics conserves the total energy given by the following Hamiltonian function
\begin{equation}\label{ham}
H_{\mathbf{G}}(\mathbf{s},\{ E_{ij} \})=\sum_{i,j \in \Lambda}A_{ij}  \left( s_i s_j + E_{ij}\right).
\end{equation}

As shown in \cite{ACV2009}, the magnetic and kinetic energies result to be non-correlated: this allows a natural definition of temperature at equilibrium, which, in a very natural way, depends only on the average kinetic energy. In fact, the link energy satisfies the Boltzmann distribution $\exp(-\beta E_{ij})$, and the fitted constant $\beta=1/T$ just corresponds to the expected inverse temperature of the system. 

Finally, we stress that the possible coupling with thermostats set at a temperature $\bar{T}$ can be realized straightforwardly by selecting a subset of links $\Gamma' \subset \Gamma$ (or, analogously a subset of nodes) and by extracting the pertaining kinetic energies according to the Boltzmann distribution, being $\beta = 1/ \bar{T}$  \cite{ACV2009}.


\section{Metrization}\label{sec:metrica}
As foresaid, an interesting characterization of the Ising model on the SG may be accomplished by a configurational analysis defined in the wider context of Cellular Automata on graphs.
Such an analysis can be realized by a particular metrization referring non directly to the configuration space, whose points are the states $\mathbf a, \mathbf b,\mathbf c,.. $ of the system, but to a peculiar {\sl partition space} containing - among other elements - the cluster distributions of the system. The mathematical framework, which  is summarized in the Appendix $A.1$, requires that the graph is endowed with the structure of a probability space. Precisely, we consider the triple $(\mathbf G,\mathcal G, \mu ) $, where the measure $\mu$ on the subset algebra $\mathcal G $ of the gasket $\mathbf G$ is simply given by the normalized number of nodes in every subset. A {\sl configuration} (or {\sl state}) on $\mathbf G$ is a function assigning to each node a value in an alphabet
$\mathbb K $. The set of all possible configurations will be denoted as $\mathcal C (\mathbf G)$, the configuration space.
Since we consider the Ising model on $\mathbf G $, the alphabet is binary, but all we are going to say is independent of the number  $ \mid \mathbb K \mid $. The adjacency matrix, combined with the list of values on nodes, allows an easy definition
of clusters on an arbitrary structure (the procedure is identical to the recognization of connected subset in graph colouring): two homogeneous nodes belong to the same cluster if they are connected through a path of nodes sharing the same value. Thus, every state $\mathbf a$ determines in a natural way  a {\sl partition}  of $\mathbf G $, i.e. an exhaustive collection $\alpha = \Phi(\mathbf a )  \equiv \{ A_1, A_2,...,A_n \} $ of disjoint subsets $ A_i$, each connected and homogeneous, commonly called {\sl atoms} of the partition. The set of all partitions of $\mathbf G$ constitutes the {\sl partition space} $\mathcal Z \equiv  \mathcal Z (\mathbf G )$. The application $ \Phi : \mathcal C (\mathbf G) \to \mathcal Z (\mathbf G )$ is many-to-one, since, for instance, permutations in $\mathbb K $ produce different states but the same partition. Obviously, $\mathcal Z (\mathbf G ) $  contains much more partitions than those derived from clusterization, e.g. since, in general, atoms do not require to be connected sets, as clusters are. In the present case,  $\mathcal Z (\mathbf G)$ is discrete and finite because $\mathbf G$ is such, but the formalism applies in abstract probability spaces (see \cite{bill,AA,rohlin,sinai,rohlin2}).

Clearly, when a dynamics is defined on the graph, this determines configuration orbits  $\{ \mathbf a(t_n) \} $ starting from any initial state $ \mathbf a(t_0) $, and the corresponding partition orbits $\{ \alpha(t_n) \}$, where $ \alpha(t_n)  \equiv  \Phi (\mathbf a(t_n) )$

Basic operations between two arbitrary partitions $\alpha$ and $\beta$  are the minimal common multiple 
$ \gamma = \alpha \vee  \beta$, and the maximal common factor (m.c.f.) or {\it intersection} $\sigma = \alpha \wedge \beta$  (see Fig.~\ref{fig:example} and Appendix $A.1$ for details).
The entropy $H({\sigma})$ of the intersection $\sigma$, when calculated between partitions at next steps along an orbit, is an index of the relevance of the (instantaneous) non evolving part.

The metrization of the partition spaces is  based on the Rohlin distance, which describes the non-similarity of two arbitrary partitions $\alpha$ and $\beta$. This distance, requiring the Shannon conditional entropy $H(\alpha | \beta) $ of measurable partitions (see  Eq.~\ref{condiz}), is given by the functional of Eq.~\ref{rohlin}, we anticipate here:
\begin{equation}
d_R (\alpha,\beta)
=H(\alpha|\beta)+H(\beta|\alpha).
\end{equation}
In order to amplify non similarity, in the Appendix $A.2$ we present also a method, referred to as ``reduction process'' and denoted $\pi$, which acts on couples of partitions and uses both operations $\vee$ and $\wedge$. More precisely, given two partitions, say $\alpha$ and $\beta$, their reduction is obtained by first defining their intersection $\sigma = \alpha \wedge \beta$ and by keeping from both partitions only those subfactors $ \widehat{\alpha}_k$ and $ \widehat{\beta}_k$, prime with $\sigma$, namely such that $\widehat{\alpha}_k \wedge \sigma = \widehat{\beta}_k \wedge \sigma = \nu$. Then, the reduced partitions are given by, $\widehat{\alpha} = \vee_j \widehat{\alpha}_j$ and $\widehat{\beta} = \vee_j \widehat{\beta}_j$ respectively (see Fig.~\ref{fig:All}, lower panel). The process $\pi$ gives evidence of the essentially different sub-partitions of any couple in $ \mathcal Z (\mathbf G )$, and therefore amplifies their distance: $d_R(\alpha ,\beta) \leq d_R(\pi(\alpha ,\beta))$. Hence, by comparing the distance between reduced and non-reduced couples, it is possible to introduce an {\sl amplification ratio}, that is $d_R( \pi[\alpha, \beta])/d_R( \alpha, \beta)$, which provides further information about the cluster distribution and mobility. However, all this analysis has to be performed in correlation with other observables. As explained in Appendix $A.3$, the reduction process is effective, namely gives rise to a large amplification ratio, whenever one of the two partitions, say $\alpha$, displays at least one cluster which in $\beta$ is exactly decomposed into smaller ones. Now, in the case under study $\alpha$ and $\beta$ are partitions defined by cluster configurations at two successive steps, and the existence of a common cluster is furthered by the special topology of the SG: clusters corresponding to subgraphs which are (combinations of) gaskets of generation $<g$ are rather stable (the border is made by two vertices only), nonetheless internal fluctuations may occur and hence decompose the cluster itself.
  
It is worth underlining that while the    $ \gamma = \alpha \vee  \beta$ operation is rather trivial, the $\sigma = \alpha \wedge \beta$ operation and the reduction process $\pi$ are quite tricky. Reduction in particular is the main algorithmical obstacle in handling large graphs ($g > 6$). 


Of course, the Rohlin distance is deeply different from the well known Hamming distance $d_H$ in the configuration space, i.e. 
\begin{equation} \label{eq:dH}
d_H (\mathbf a,\mathbf b) = {1\over N}\sum_1^{N} \rho (a_i,b_i),
\end{equation}
where $a_i$ and $b_i$ are the values of the $i-$th node, $\rho$ is a distance functional in the alphabet. The simplest distance in $ \mathbb K $ is $\rho(x,y)= 1 - \delta_{xy}$, leading to Hamming distance 0.5 for purely random configurations. Looking for instance to Fig.~\ref{fig:All} (upper panel), we would get the maximal Hamming distance for the configurations $\mathbf{a}$ and $\mathbf{b}$,  and a minor distance for  $\mathbf{a}$ and $\mathbf{c}$,  while the Rohlin metrics on the corresponding partitions gives null distance in the former case and a high distance in the latter. 
In other terms, $d_R$ and $d_H$ are  deeply different in principle
not only because they refer to different objects (defined in  $\mathcal Z(\mathbf G) $ and $\mathcal C (\mathbf G)$, respectively), but also because the former has do to with mutual distribution of clusters, which could involve geometrical features and long range correlations, while the latter is the sum of strictly local differences. Therefore, even if the correspondence $\Phi : \mathcal C \to \mathcal Z $ is many-to-one, the loss of information should be compensated by the fact that by $d_R$ we get a global estimate on cluster distributions, instead of a bare sum of uncorrelated differences.
Finally, more details concerning the implementation of the algorithm for measure of the above mentioned metric observables can be found in Appendix $A.4$.





\section{Numerical Experiments and Results}\label{sec:numerics}

We now focus on the Ising model on the SG, with the microcanonical dynamics described in Sec.~\ref{sec:Micro}. 
The equilibrium regime is ensured  by coupling the system to thermostats set at the same  temperature; in this particular case the simplest way is to consider, for a given generation $g$, the external triangle, i.e the $3 \times 2^{g-1}$ links defining the perimeter of the gasket; if then we exclude the six angular links, we get $3$ separated thermostats. 
This could be useful in the future, as it consistutes a quite simple way of imposing temperature gradients to the system. However, it should be noticed that, due to the fact that the contact between the thermostats and the system gets vanishingly small with respect to the bulk as $g$ is increased, the time for thermalization is expected to grow with the size of the system. Anyway, as mentioned, we are now interested only in the equilibrium behavior.

Before proceeding, we underline that our checks strongly confirm that the thermalized states reached by the microcanonical dynamics are consistent with those expected from a canonical dynamics, e.g. based on the Metropolis algorithm. In particular, we verified that, for a given temperature, macroscopic observables like the magnetization and the energy measured with the two kinds of dynamics are indistinguishable.  

For any given temperature $T$, the geometric observables have been calculated as time series starting after a thermalization time, and lasting an observation time $t_{\mathrm{max}}$, where time is measured in units of Monte Carlo (MC) steps; i.e. $N$ elementary moves.  
In particular, we consider finite segments of trajectories $\{ \mathbf a(t_k) \} $ in $\mathcal C$ or $\{ \alpha(t_k) \equiv \Phi (\mathbf a(t_k)  )\} $ in $\mathcal Z$, for $t_k = 0,1,2,...,t_{\mathrm{max}}$, as well as the related
intersections  $\sigma(t_k) = \alpha(t_k) \wedge \alpha(t_{k+1})$ and the couples of reduced partitions at successive time steps
$(\widehat \alpha ,\widehat \beta ) = \pi(\alpha ,\beta ) $.
From such equilibrium trajectories we obtain segments of time series for the following  quantities:

\begin{enumerate}
\item the entropy $H(\cdot)$, applied either to the orbits $\{ \alpha(t_k)\}$ or $\{ \sigma(t_k)\}$;
\item the Rohlin distance $d_R(t_k) \equiv  d_R( \alpha(t_{k-1}), \alpha(t_k))$, (see Eq.~\ref{rohlin});
\item the amplified Rohlin distance $ \widehat{d_R}(t_k) \equiv d_R( \pi[\alpha(t_{k-1}), \alpha(t_k)])$;
\item the amplification ratio $\widehat{d_R}(t_k)  / d_R(t_k)$;
\item the Hamming distance $d_H(t_k) = d_H(\mathbf{a}(t_{k-1}),\mathbf{a}(t_k))$, (see Eqs.~\ref{eq:dH} and \ref{hamming}).
\end{enumerate}

Here we focus to the case of unitary time steps ($t_{k+1}- t_k$ equals one MC step), leaving the study of the role of the time gap to future investigations.

After setting the experimental parameters (size, thermalization time, confidence length of trajectories $t_{\mathrm{max}}$), we calculate time averages, variances etc. for each of the series above. Results found for different choices of size (we especially focused on gaskets of generation $g=4$, $g=5$ and $g=6$, corresponding to $N=120 $, $N=366$ and $N=1095$ sites respectively) and of $t_{\mathrm{max}}$ are qualitatively in very good agreement. Moreover, to approach the thermalized state, the initial configuration is taken ferromagnetic; this minimizes the likelihood of pinning effects during the evolution \cite{ale}. In the following we report only the essential information, dropping redundant numerical outputs.  


In general, the observables analyzed highlight the existence of different regimes, demarcated by remarkable temperatures $T_I\approx 1.3$ and $T_{II} \approx 2.0$. More precisely, $T_I$ corresponds to 
the flex in the Rohlin distance and to the peak in the amplification ratio, while $T_{II}$ corresponds to the peak in the variance of Rohlin distance and to the crossover in the intersection entropy (see Figs.~$2-4$). Interestingly, $T_I$ and  $T_{II}$ recover the ``critical'' temperatures $T_1$ and $T_2$ 
evidenced by thermodynamic analysis (see Sec.~$2$ and \cite{gefen1,gefen2,liu}). Indeed, consistently with thermodynamic results, the highlighted regimes correspond to a long-range order region and to a disordered  region with a critical-like transition region in between. More precisely:  

\begin{figure}\label{fig:d_r}
\includegraphics[width=1.0\textwidth]{Rohlin.eps}
\caption{(Color on line) Mean (left panel)  and variance (right panel) for Rohlin's distance $d_R$ as a function of the temperature; three different sizes are compared.}
\end{figure}

\begin{figure}\label{fig:d_h}
\includegraphics[width=1.0\textwidth]{Hamming.eps}
\caption{(Color on line) Mean (left panel)  and variance (right panel) for Hamming's distance $d_H$ as a function of the temperature; three different sizes are compared.}
\end{figure}





\begin{itemize}
\item for $ T < T_{I} $, $\langle m^2 \rangle$ is close to $1$ (see \cite{liu}), while the distances $d_H$ and $d_R$ are approximately $0$ (see Figs.~$2$ and $3$); in fact, at such small temperatures a ferromagnetic order is established over large length-scales: clusters are constituted by few single spots in the large ``sea'' of equally oriented spins and  spin-flips are rather unlikely to happen. So, configurations - and partitions - at consecutive steps
differ for such small spots that distances are  extremely small. Moreover, during a MC step, the rare spin flips occurring are yet able to change the atoms in such a way that overlaps between consecutive partitions are quite improbable. This inhibits the reduction, and the amplification is close to 1. 
For $T$ approaching $T_I$,  such spots get larger, but, due to the slowness of the evolution at low temperature, their borders can remain sufficiently unchanged for several steps. Hence, the emergence of spots within such clusters allows the reduction, processing couples of consecutive partitions, to get more effective, and the time average of the amplification is manifestly increased. The growth of the spots can be retrieved by the intersection entropy increase, as shown in the inset of Fig.~$4$ (left panel). 

\item for $T \approx T_I$, $\langle m^2 \rangle$ is close to $0.5$ and both $d_H$ and $d_R$ start to be significantly larger than $0$. This  is a consequence of the fact that spin flips are getting more frequent. More interestingly, the amplification ratio reaches a maximum. 
We have seen that, at the middle of the previous regime, two conditions cooperated to start the growth of the amplification ratio: non empty intersection $\sigma$, and the fact that at consecutive steps there exist uniform large clusters which are decomposed internally, yielding an effective reduction (as explained in Appendix $A.3$). Here, the peak in the amplification ratio means that cluster sizes and the speed of the dynamics optimally fit such conditions; moreover, fixed clusters are more likely. The exact determination of the peak temperature is tricky, due to the growing complexity of the configuration dynamics, but we argue that $T_I$ 
(or $T_1$) is a good approximation. This behavior is consistent with the apparent transition occurring at $T_1$ as a long-range order breakdown. 


\item for $T_I < T < T_{II}$,  $\langle m^2 \rangle$ is approaching zero. Clusters get more intricate due to the growing temperature, and  $H({\sigma})$, which measures the relevance of overlapping between successive configurations, exhibits a rapid growth (see Fig.~$4$, left panel). The coexistence of fragmentation  of large clusters, which is a signature of decay for long range correlations, and overlapping may be related to ``critical slowing down'' effects \cite{CSD}.  Clearly, fragmented clusters have a higher probability to overlap, but are unlikely to include small spots; As a consequence the amplification ratio is still larger than $1$, though rapidly decreasing. 
Note that $H({\sigma})$ is rather  far from saturation (namely, uncorrelated chaos) and this suggests the persistence of  
a local short-length order. 
\item for $ T \sim T_{II}$, the amplification ratio is practically 1: such a breakdown of the reduction, as forementioned,  has a completely different meaning than in the case of very low temperatures, where overlapping was insignificant for the smallness of
sparse clusters in the ``big sea'' of dominant magnetic orientation. Here, due to the smallness of the $\sigma$ atoms, it is unlikely that a sufficiently large cluster is internally decomposed at the next step (see also the Appendix $A.3$). 

At this temperature the variance of $d_R$ has a maximum (whose value scales like $1/\sqrt{N}$), and this constitutes a signature that, along the trajectory, the fluctuation  of the distance is particularly important. Indeed, the complexity of clusters shape and their mobility can give rise to wide fluctuations of distances in time. 
Also, $H({\sigma})$ exhibits a crossover: from $T_{II}$, the growth is due only to fragmentation, while the ``critical
slowing down'' is over. Both the persistent growth of $H({\sigma})$ and the value of $d_H $ (neatly below $0.5$) indicate that the complete chaos is far from being established at this
temperature and for these sizes. Moreover, the large fluctuations make the system more susceptible to spin-flips, i.e. energy changes, and this is consistent with the peak in the specific heat.

\item for $T > T_{II}$ , the Rohlin variance decreases roughly as $\sim 1/N$ toward an asymptotic value corresponding to the uncorrelated chaos, whose onset may be recognized by $d_H \simeq 0.5$. We argue that the fragmentation and the mobility of clusters stabilize the behavior of the time series for global quantities like distances.


\begin{figure}\label{fig:ampl}
\includegraphics[width=1.0\textwidth]{Hsigma2.eps}
\caption{(Color on line) Intersection entropy $H(\sigma)$ (left panel) and amplification ratio $\widetilde{d_R}/d_R$ (right panel) as a function of the temperature. For the latter, accurate numerical data are presented only for generation $g=5$, due to time consuming calculations; nonetheless we have checked that for other values of $g$ the qualitative behavior is robust. The inset of the left panel shows a magnification of the departure from zero of the entropy; notice the semilogarithmic scale.}
\end{figure}

We note however that the maximum for the $d_H$-variance occurs at $T\approx 3$, well above the corresponding maximum of the
$d_R$-variance at $T\approx 2$. We deduce that in this temperature interval big clusters of all sizes, typical of the previous fractal configurations, have started their fragmentation, so that the global shape diversity between subsequent configurations is 
 stabilizing and this explains why the $d_R$-variance decreases. However, the fragmentation process is still slow enough that the local matching between different or equal spins at next steps is highly unstable in time, allowing for the growth of the $d_H$-variance. Indeed, spins belonging to the inner part of a cluster (even if of small size)  result from the dynamics to be more stable than those belonging to its periphery. For $T> 3$ the fragmentation is such that both local overlaps and medium length correlations tend to stabilize the time behavior. There is therefore a temperature interval exhibiting a subtle interplay between correlation length and time stability. In other terms, in the way to the chaos we recognize two time scales: one in terms of global similitude (shape) of clusters along the orbits, which starts to stabilize just at the end of the ``critical slowing down''
($T\approx 2$); the other in terms of local overlaps, whose stabilization requires a higher temperature  ($T\approx 3$). The direct visual inspection could be awkward for such properties, undetectable also from the mean magnetization.

\end{itemize} 

\section{Conclusions and perspectives}\label{sec:con}

In this work we have performed a quantitative characterization of cluster dynamics for the Ising model defined on a finite Sierspinski Gasket, whose thermodynamic behavior is known to be non-trivial.
The analysis is based on a set of geometric observables, such as Hamming and Rohlin distances, and on reduction operations among partitions which allow to detect the evolving and the stable components of clusters.

The phenomenology evidenced by previous thermodynamic analysis is qualitatively and quantitatively confirmed by the present metric approach, which provides in addition a geometric characterization of the anomalous behavior of the system. Indeed, we highlight first the existence of two ``critical'' temperatures: $T_I$ corresponding to a peak in the amplification ratio, meaning that
there an abrupt change in the clusters behavior occurs, and $T_{II}$, corresponding to a peak in 
the Rohlin variance and other crossovers, evidencing the loss of short range order.  
More precisely, the amplification ratio gives detailed information about the development of clusters, starting
from little independent spots (amplification ratio equal to 1) to first overlaps (amplification ratio larger than 1 and growing to
the maximum) while the next phase of decreasing ratio indicates the complex effect of decreasing efficiency of the reduction
due to the fragmentation, up to $T_{II}$ when the fragmentation is such that,
notewithstanding the large intersection, the reduction process is inhibited by the extreme improbability of clusters with a decomposable inner part (see  \ref{ampli}). 


Another new information we obtain in the high temperature regime is the existence of  different scales in the destruction of local and correlated order, as evidenced by the Hamming and Rohlin variances, with an anticipated peak for the latter quantity.



Thus, the analysis above enabled us to distinguish different behaviors and phases demarcated by $T_I$ and $T_{II}$ as main milestones; we cannot speak of course of ``phase transitions'', since the onset of these distinct behaviors seems to be a rather smooth process. However, from such a detailed description of the collective motion we get an insight on the interplay between dynamic and statistical features, especially the decay of spatial and temporal correlations.

This work opens a lot of possible extensions and further insights. 
The next step is the metric characterization of clusters dynamics in condition of non-equilibrium; 
a kind of analysis allowed by the chosen dynamics even in the presence of a disordered coupling pattern \cite{ACV2009}. 

In particular, it could be interesting to deepen the effect of the double scale in the approach to uncorrelated chaos on the conductivity. In the same context, another interesting item to explore is the relevance of topology for the cluster diffusion. Indeed, the whole set of operations performed on the SG, from cluster identification to  reduction, immediately applies to any automaton on arbitrary, connected graphs: the process depends only on the adjacency matrix, on a generic alphabet and on a dynamics, working as a proper external engine generating a succession of configurations. 
Therefore, by the bare substitution of the adjacency matrix, the algorithm is ready to fit a great variety of statistical models (e.g. Pott's model on lattices of arbitrary dimension or graphs), or other network problems where a dynamics is defined. If necessary, nodes could also be weighted, defining alternative probability measures. 
True algorithmic problems and non trivial extensions could only arise from alternative definitions of partitions (assuming e.g. atoms of a different kind with respect to the clusters) or from a different factorization, modifying the reduction process. 





