Uniqueness of the Gibbs measure for the $4$-state anti-ferromagnetic Potts model on the regular tree
David de Boer, Pjotr Buys, Guus Regts
备注：
Subsection 1.2 and Section have been merged and slightly rewritten. Proof of Lemma 1.3 has been moved to an appendix, fixed a small mistake in the proof of this lemma. Some small other changes have been made. No significant changes. Accepted in CPC
We show that the $4$-state anti-ferromagnetic Potts model with interaction parameter $w\in(0,1)$ on the infinite $(d+1)$-regular tree has a unique Gibbs measure if $w\geq 1-\frac{4}{d+1}$ for all $d\geq 4$. This is tight since it is known that there are multiple Gibbs measures when $0\leq w<1-\frac{4}{d+1}$ and $d\geq 4$.
We moreover give a new proof of the uniqueness of the Gibbs measure for the $3$-state Potts model on the $(d+1)$-regular tree for $w\geq 1-\frac{3}{d+1}$ when $d\geq 3$ and for $w\in (0,1)$ when $d=2$.
We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e on a complete graph) and on trees. For the mean field case, a complete characterization of the set of temperatures for which non-Gibbsianness happens is given. The results for trees are somewhat less explicit, but we do show for general trees that non-Gibbsianness of the fuzzy Potts model happens exactly for those temperatures where the underlying Potts model has multiple Gibbs measures.
In this paper, we consider the Potts-SOS model where the spin takes values in the set $\{0, 1, 2\}$ on the Cayley tree of order two. We describe all the translation-invariant splitting Gibbs measures for this model in some conditions. Moreover, we investigate whether these Gibbs measures are extremal or non-extremal in the set of all Gibbs measures.
In this paper we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology etc.
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…
We consider the Curie-Weiss Potts model in zero external field under independent symmetric spin-flip dynamics. We investigate dynamical Gibbs-non-Gibbs transitions for a range of initial inverse temperatures beta<3, which covers the phase transition point beta=4 log 2 [8]. We show that finitely many types of trajectories of bad empirical measures appear, depending on the parameter beta, with a possibility of re-entrance into the Gibbsian regime, of which we provide a full description.
We prove that all Gibbs states of the q-state nearest neighbor Potts model on Z^2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the i…
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.
The antiferromagnetic $q$-state Potts model is perhaps the most canonical model for which the uniqueness threshold on the tree is not yet understood, largely because of the absence of monotonicities. Jonasson established the uniqueness threshold in the zero-temperature case, which corresponds to the $q$-colourings model. In the permissive case (where the temperature is positive), the Potts model has an extra parameter $β\in(0,1)$, which makes the task of analysing the uniqueness threshold even harder and much less is known. In this paper, we focus on the case $q=3$ and give a detailed analysis…
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 …
In the present paper, we study a phase transition problem for the $q$-state $p$-adic Potts model over the Cayley tree of order three. We consider a more general notion of $p$-adic Gibbs measure which depends on parameter $ρ\in\bq_p$. Such a measure is called {\it generalized $p$-adic quasi Gibbs measure}. When $ρ$ equals to $p$-adic exponent, then it coincides with the $p$-adic Gibbs measure. When $ρ=p$, then it coincides with $p$-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of $|ρ|_p$. Namely, in the first regime, one takes $ρ=\exp_p(J)$ for some $…
The paper concerns the $q$-state Potts model (i.e., with spin values in $\{1,\dots,q\}$) on a Cayley tree $\mathbb{T}^k$ of degree $k\geq 2$ (i.e., with $k+1$ edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invari…
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 paper, we prove the existence of infinite Gibbs Delaunay Potts tessellations for marked particle configurations. The particle systems has two types of interaction, a so-called \emph{background potential} ensures that small and large triangles are excluded in the Delaunay tessellation, and is similar to the so-called hardcore potential introduced in \cite{Der08}. Particles carry one of $q$ separate marks. Our main result is that for large activities and high \emph{type interaction} strength the model has at least $ q$ distinct translation-invariant Gibbs Delaunay Potts tessellations. Th…
In this paper we study a continuum version of the Potts model. Particles are points in R^d, with a spin which may take S possible values, S being at least 3. Particles with different spins repel each other via a Kac pair potential. In mean field, for any inverse temperature there is a value of the chemical potential at which S+1 distinct phases coexist. For each mean field pure phase, we introduce a restricted ensemble which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin Shlosman theory, we get uniqueness and expone…
We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q-ary channels. The bounds are optimal for the Ising model and appear to be close to what we conjecture to be the true values up to a factor of 0.0150 in the case q = 3 and 0.0365 for q = 4. Our proof uses an iteration …
We give a detailed and refined proof of the Dobrushin-Pechersky uniqueness criterion extended to the case of Gibbs fields on general graphs and single-spin spaces, which in particular need not be locally compact. The exponential decay of correlations under the uniqueness condition has also been established.
We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers $q\geq 3$ and $Δ\geq 3$, we develop algorithms that produce samples within error $o(1)$ from the $q$-state Potts model on random $Δ$-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic …
We present a proof of the power law decay of magnetic moment for the $q$-state antiferromagnetic Potts model on the regular tree at the critical temperature, and also justify that the exact exponent is $\frac{1}{2}$. Our proof relies on the assumption of the uniqueness at the critical temperature, which has been established for $q=3,4$, and for $q \ge 5$ with large degree. An iterative contraction inequality is developed for independent interests.
We consider Gibbs measures relative to Brownian motion of Feynman-Kac type, with single site potential V. We show that for a large class of V, including the Coulomb potential, there exist infinitely many infinite volume Gibbs measures.
We study equilibrium states of an infinite system of interacting particles in a Euclidean space. The particles bear `unbounded' spins with a given symmetric a priori distribution. The interaction between the particles is pairwise and splits into position-position and spin-spin parts. The position-position part is described by a superstable potential, and the spin-spin part is attractive and of finite range. Thermodynamic states of the system are defined as tempered Gibbs measures on the space of marked configurations. We derive sufficient conditions of the existence and uniqueness of these Gib…
We define a potential-weighted connective constant that measures the effective strength of a repulsive pair potential of a Gibbs point process modulated by the geometry of the underlying space. We then show that this definition leads to improved bounds for Gibbs uniqueness for all non-trivial repulsive pair potentials on $\mathbb R^d$ and other metric measure spaces. We do this by constructing a tree-branching collection of densities associated to the point process that captures the interplay between the potential and the geometry of the space. When the activity is small as a function of the p…
We extend results on quadratic pressure and convergence of Gibbs mesures from previous joined work of the authors to the Curie-Weiss-Potts model. We define the notion of equilibrium state for the quadratic pressure and show that under some conditions on the maxima for some auxiliary function, the Gibbs measure converges to a convex combination of eigen-measures for the Transfer Operator. This extension works for dynamical systems defined by {infinite-to-one} maps. As an example, we compute the equilibrium for {the mean-field} $XY$ model as the number of particles goes to $+\infty$.
We develop a new pressure representation theorem for nearest-neighbour Gibbs interactions and apply this to obtain the existence of efficient algorithms for approximating the pressure in the $2$-dimensional ferromagnetic Potts, multi-type Widom-Rowlinson and hard-core models. For Potts, our results apply to every inverse temperature but the critical. For Widom-Rowlinson and hard-core, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of our work is in the latter.
We prove that for q>=1, there exists r(q)<1 such that for p>r(q), the number of points in large boxes which belongs to the infinite cluster has a normal central limit behaviour under the random cluster measure phi_{p,q} on Z^d, d>=2. Particularly, we can take r(q)=p_g^* for d=2, which is commonly conjectured to be equal to p_c. These results are used to prove a q-dimensional central limit theorems relative to the fluctuation of the empirical measures for the ground Gibbs measures of the q-state Potts model at very low temperature and the Gibbs measures which reside in the convex hull of them. …
We present general existence and uniqueness results for marked models with pair interactions, exemplified through Gibbs point processes on path space. More precisely, we study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to $\mathbb{R}^d$, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.
We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity $z$ and inverse temperature $β$. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interactions.
In in this paper we establish an explicit and sharp estimate of the spectral gap (Poincaré inequality) and the transportation inequality for Gibbs measures, under the Dobrushin uniqueness condition. Moreover, we give a generalization of the Liggett's $M-ε$ theorem for interacting particle systems.
We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show the equivalence of correlations of the grand canonical (gce) and the canonical ensemble (ce). As a corollary we obtain that the correlations of the ce decay exponentially plus a volume correction term. Then, we use the decay of correlation to verify a conjecture that the infinite-volume Gibbs measure of the ce is unique on a one-dimensional lattice. For the equivalence of correlations, we modify a method that was recently used by the authors to show t…
In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box $[-n,n]^d$ (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sob…
We study the performance of Markov chains for the $q$-state ferromagnetic Potts model on random regular graphs. It is conjectured that their performance is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these pha…
We study single-site stochastic and deterministic transforma- tions of one-dimensional Gibbs measures in the uniqueness regime with infinite-range interactions. We prove conservation of Gibbsianness and give quantitative estimates on the decay of the transformed potential. As examples, we consider exponentially decaying potentials, and potentials decaying as a power-law.
We consider - in uniformly strictly convex potential regime - two versions of random gradient models with disorder. In model (A) the interface feels a bulk term of random fields while in model (B) the disorder enters though the potential acting on the gradients. We assume a general distribution on the disorder with uniformly-bounded finite second moments. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension $d = 2$, while there are shift-invariant gradient Gibbs measures describing an infinite-volume distribution for the gradien…
Uniqueness, analyticity and mixing for Gibbs point processes via spectral gaps
Hasso Plattner Institute, University of Potsdam（波茨坦大学哈索·普拉特纳研究所）；King’s College London, Department of Mathematics（伦敦国王学院数学系）；Northwestern University, Department of Mathematics（西北大学数学系）；Georgia Institute of Technology, School of Computer Science（佐治亚理工学院计算机学院）
We study Gibbsian models of unbounded integer-valued spins on trees which possess a symmetry under height-shift. We develop a theory relating boundary laws to gradient Gibbs measures, which applies also in cases where the corresponding Gibbs measures do not exist. Our results extend the classical theory of Zachary beyond the case of normalizable boundary laws, which implies existence of Gibbs measures, to periodic boundary laws. We provide a construction for classes of tree-automorphism invariant gradient Gibbs measures in terms of mixtures of pinned measures, whose marginals to infinite paths…
We consider gradient fields on $\mathbb{Z}^d$ for potentials $V$ that can be expressed as $$e^{-V(x)}=pe^{-\frac{qx^2}{2}}+(1-p)e^{-\frac{x^2}{2}}.$$ This representation allows us to associate a random conductance type model to the gradient fields with zero tilt. We investigate this random conductance model and prove correlation inequalities, duality properties, and uniqueness of the Gibbs measure in certain regimes. Moreover, we show that there is a close relation between Gibbs measures of the random conductance model and gradient Gibbs measures with zero tilt for the potential $V$. Based on …
The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of its updates. We present optimal bounds on the convergence rate of the Swendsen-Wang algorithm for the complete $d$-ary tree. Our bounds extend to the non-uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known as Variance Mixing and Entropy Mixing, introduced in the study of local Markov chains by Martinelli e…
We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models sub jected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations. We exhibit the minimal necessary structure for such double-layer systems. Assuming no a priori metric on the local state spaces, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by …
We present novel results for fast mixing of Glauber dynamics using the newly introduced and powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. In our results, the parameters of the Gibbs distribution are expressed in terms of the spectral radius of the adjacency matrix of $G$, or that of the Hashimoto non-backtracking matrix. The analysis relies on new techniques that we introduce to bound the maximum eigenvalue of the pairwise influence matrix $\mathcal{I}^{Λ,τ}_{G}$ for the two spin Gibbs distribution $μ$. There is a common framework that underlies these techni…
The Dobrushin comparison theorem is a powerful tool to bound the difference between the marginals of high-dimensional probability distributions in terms of their local specifications. Originally introduced to prove uniqueness and decay of correlations of Gibbs measures, it has been widely used in statistical mechanics as well as in the analysis of algorithms on random fields and interacting Markov chains. However, the classical comparison theorem requires validity of the Dobrushin uniqueness criterion, essentially restricting its applicability in most models to a small subset of the natural pa…
We discuss the relationship between discrete-time processes (chains) and one-dimensional Gibbs measures. We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specific…
Kittel's 1D model represents a natural DNA with two strands as a (molecular) zipper, which may separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a non-linear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measu…
A lattice system of interacting temperature loops, which is used in the Euclidean approach to describe equilibrium thermodynamic properties of an infinite system of interacting quantum particles performing anharmonic oscillations (quantum anharmonic crystal), is considered. For this system, it is proven that: (a) the set of tempered Gibbs measures is non-void and weakly compact; (b) every Gibbs measure obeys an exponential integrability estimate, the same for all such measures; (c) every Gibbs measure has a Lebowitz-Presutti type support; (d) the set of all Gibbs measures is a singleton at hig…
We formulate a continuous version of the well known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $λ> 0$. In this version, the state or "spin value" $x_u$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_u + x_v \leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $λ>0$. Our result shows that…
