We consider the ferromagnetic q-state Potts model with zero external field in a finite volume evolving according to Glauber-type dynamics described by the Metropolis algorithm in the low temperature asymptotic limit. Our analysis concerns the multi-spin system that has q stable equilibria. Focusing on grid graphs with periodic boundary conditions, we study the tunneling between two stable states and from one stable state to the set of all other stable states. In both cases we identify the set of gates for the transition and prove that this set has to be crossed with high probability during the transition. Moreover, we identify the tube of typical paths and prove that the probability to deviate from it during the transition is exponentially small.
In this article we create a new algorithm for the perfect simulation of the infinite Potts model at a sufficiently small or at a sufficiently high temperature, in particular under the transition phase temperature. We study the model for free boundary conditions and we give some consequences for the constant boundary conditions.
This work is concerned with the theory of Graphical Representation for the Ising and Potts Models over general lattices with non-translation invariant external field. We explicitly describe in terms of the Random Cluster Representation the distribution function and, consequently, the expected value of a single spin for the Ising and $q$-states Potts Models with general external fields. We also consider the Gibbs States for the Edwards-Sokal Representation of the Potts Model with non-translation invariant magnetic field and prove a version of the FKG Inequality for the so called General Random …
We present a new and simple proof for the classic results of Imbrie (1985) and Bricmont-Kupiainen (1988) that for the random field Ising model in dimension three and above there is long range order at low temperatures with presence of weak disorder. With the same method, we obtain a couple of new results: (1) we prove that long range order exists for the random field Potts model at low temperatures with presence of weak disorder in dimension three and above; (2) we obtain a lower bound on the correlation length for the random field Ising model at low temperatures in dimension two (which matche…
Motivated by recent developments over the past few years in the study of the correlation length of the random-field Ising model due to Ding and Wirth in a paper first available in 2020, we pursue one natural direction of research that the authors propose is of interest, namely in confirming that the same scaling for the correlation length for the random-field Ising model equals that of the random-field Potts model. To demonstrate that the $\frac{4}{3}$ emergence in the correlation length scaling for the random-field Potts model coincides with that correlation length scaling for the random-fiel…
Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensiona…
In this study, we investigate the energy landscape of the Ising and Potts models on fixed and finite but large three-dimensional (3D) lattices where no external field exists and quantitatively characterize the metastable behavior of the associated Glauber dynamics in the very low temperature regime. Such analyses for the models with non-zero external magnetic fields have been extensively performed over the past two decades; however, models without external fields remained uninvestigated. Recently, the corresponding investigation has been conducted for the two-dimensional (2D) model without an …
We consider the Potts model with $q$ colors on a sequence of weighted graphs with adjacency matrices $A_n$, allowing for both positive and negative weights. Under a mild regularity condition the mean-field prediction for the log partition function of the Potts model on a sequence of matrices $A_n$ is asymptotically correct, whenever $\text{tr}(A_n^2)=o(n)$. In particular, our results are applicable for the Ising and the Potts models on any sequence of graphs with average degree going to $+\infty$. Using this, we establish the universality of the limiting log partition function of the ferromagn…
We complete the investigation of the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction which was started by Jahnel and one of the authors. As our main result of the present paper, we extend the previous sharpness result of mean-field bounds to cover all possible cases of fuzzy transformations, allowing also for the occurrence of Ising classes. The closing of this previously left open Ising-gap involves an analytical argument showing uniqueness of minimizing profiles for certain non-homogeneous conditional variational problems.
We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree $d$, $\Td$. For given interaction parameters $J_1$, $J_2>0$ and external field $h_1\in\RR$, we compute the smallest external field $\tilde{h}$ such that the plus measure with parameters $J_2$ and $h$ dominates the plus measure with parameters $J_1$ and $h_1$ for all $h\geq\tilde{h}$. Moreover, we discuss continuity of $\tilde{h}$ with respect to the three parameters $J_1$, $J_2$, $h$ and also how the plus measures a…
This is a short account of connections between the Tutte polynomial and the Ising, Potts, and random-cluster models. The four principal elements are the Ising model of 1925, the Tutte polynomial of 1947, the Potts model of 1952, and the random-cluster model of 1972.
We consider the ferromagnetic $q$-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature $β$. Our analysis concerns the low-temperature regime $β\to \infty$, in which this multi-spin system has $q$ stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the $q$-state Potts model. More specifically, we describe the asymptotic behavior o…
The main contribution of the current study is two-fold. First, we investigate the energy landscape of the Ising and Potts models on finite two-dimensional lattices without external fields in the low temperature regime. The complete analysis of the energy landscape of these models was unknown because of its complicated plateau saddle structure between the ground states. We characterize this structure completely in terms of a random walk on the set of sub-trees of a ladder graph. Second, we provide a considerable simplification of the well-known potential-theoretic approach to metastability. In …
We study graphical representations for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960's. The second model is the space-time percolation process, which is closely related to the contact model for the spread of disease. We consider a `space-time' random-cluster model and explore a range of useful probabilistic techniques for studying it. The space-time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase t…
We study dynamical aspects of the $q$-state Potts model on an $n\times n$ box at its critical $β_c(q)$. Heat-bath Glauber dynamics and cluster dynamics such as Swendsen--Wang (that circumvent low-temperature bottlenecks) are all expected to undergo "critical slowdowns" in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is $O(1)$, should at criticality be polynomial in $n$ for $1< q \leq 4$, and exponential in $n$ for $q>4$ in accordance with the predicted discontinuous phase transition. This was confirmed for $q=2$ (the Ising model) by th…
We review what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and in particular we prove that the mixture $\frac12(μ^\pm+μ^\mp)$ of two reflection-symmetric Dobrushin states of the 3-dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finite-volume measures with deterministic boundary conditions. Finally we point out what the issues are in order to extend the analysis to the Potts model, and give a few conjectures.
In this article, we investigate the energy landscape and metastable behavior of the Ising and Potts models on two-dimensional square or hexagonal lattices in the low temperature regime, especially in the absence of an external magnetic field. The energy landscape of these models without an external field is known to have a huge and complex saddle structure between ground states. In the small volume regime where the lattice is finite and fixed, the aforementioned complicated saddle structure has been successfully analyzed in [20] for two or three dimensional square lattices when the inverse tem…
In this note we study the block spin mean-field Potts model, in which the spins are divided into $s$ blocks and can take $q\ge 2$ different values (colors). Each block is allowed to contain a different proportion of vertices and behaves itself like a mean-field Ising/Potts model which also interacts with other blocks according to different temperatures. Of particular interest is the behavior of the magnetization, which counts the number of colors appearing in the distinct blocks. We prove central limit theorems for the magnetization in the generalized high temperature regime and provide a mode…
Pennsylvania State University（宾夕法尼亚州立大学）；Georgia Institute of Technology（佐治亚理工学院）；University of Rochester（罗切斯特大学）
We study identity testing for restricted Boltzmann machines (RBMs), and more generally for undirected graphical models. Given sample access to the Gibbs distribution corresponding to an unknown or hidden model $M^*$ and given an explicit model $M$, can we distinguish if either $M = M^*$ or if they are (statistically) far apart? Daskalakis et al. (2018) presented a polynomial-time algorithm for identity testing for the ferromagnetic (attractive) Ising model. In contrast, for the antiferromagnetic (repulsive) Ising model, Bezáková et al. (2019) proved that unless $RP=NP$ there is no identity tes…
We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernandez, den Hollander and Martınez for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model with class size unequal two. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and u…
