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 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 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…
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 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 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 analyze the generalized mean-field q-state Potts model which is obtained by replacing the usual quadratic interaction function in the mean-field Hamiltonian by a higher power z. We first prove a generalization of the known limit result for the empirical magnetization vector of Ellis and Wang which shows that in the right parameter regime, the first-order phase-transition persists. Next we turn to the corresponding generalized fuzzy Potts model which is obtained by decomposing the set of the q possible spin-values into 1 < s < q classes and identifying the spins within these classes. In exte…
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 …
Multi-point functions of a full-plane two-state fuzzy $4$-Potts model
NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai（纽约大学-华东师范大学数学科学研究所，上海纽约大学）；Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉科学研究院）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉科学研究院）
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…
A fractional fuzzy Potts measure is a probability distribution on spin configurations of a finite graph $G$ obtained in two steps: first a subgraph of $G$ is chosen according to a random cluster measure $ϕ_{p,q}$, and then a spin ($\pm1$) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever $q\geq1$, such a measure is positively associated, meaning that any two increasing events are positively correlated. This generalizes earlier results of Häggström [Ann. Appl. Probab. 9 (1999) 1149--1159] and Häggström and Schramm [S…
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…
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 …
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 $…
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 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 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 consider $\mathbb Z$-valued $p$-SOS-models with nearest neighbor interactions of the form $|ω_v-ω_w|^p$, and finite-spin ferromagnetic models on regular trees. This includes the classical SOS-model, the discrete Gaussian model and the Potts model. We exhibit a family of extremal inhomogeneous (i.e. tree automorphism non-invariant) Gibbs measures arising as low temperature perturbations of ground states (local energy minimizers), which have a sparse enough set of broken bonds together with uniformly bounded increments along them. These low temperature states in general do not possess any sym…
We give a criterion of the form Q(d)c(M)<1 for the non-reconstructability of tree-indexed q-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix M. Here c(M) is an explicit function, which is convex over the set of M's with a given invariant distribution, that is defined in terms of a (q-1)-dimensional variational problem over symmetric entropies. Further Q(d) is the expected number of offspring on the Galton-Watson tree. This result is equivalent to proving the extremality of the free boundary condition-Gibbs measure within the corresponding Gibbs…
We study several statistical mechanical models on a general tree. Particular attention is devoted to the classical Heisenberg models, where the state space is the d-dimensional unit sphere and the interactions are proportional to the cosines of the angles between neighboring spins. The phenomenon of interest here is the classification of phase transition (non-uniqueness of the Gibbs state) according to whether it is robust. In many cases, including all of the Heisenberg and Potts models, occurrence of robust phase transition is determined by the geometry (branching number) of the tree in a way…
Pemantle and Steif provided a sharp threshold for the existence of a RPT (robust phase transition) for the continuous rotator model and the Potts model in terms of the branching number and the second eigenvalue of the transfer operator, where a robust phase transition is said to occur if an arbitrarily weak coupling with symmetry-breaking boundary conditions suffices to induce symmetry breaking in the bulk. They further showed that for the Potts model RPT occurs at a different threshold than PT (phase transition in the sense of multiple Gibbs measures), and conjectured that RPT and PT should o…
The Swendsen-Wang dynamics is a Markov chain widely used by physicists to sample from the Boltzmann-Gibbs distribution of the Ising model. Cooper, Dyer, Frieze and Rue proved that on the complete graph K_n the mixing time of the chain is at most O(n^{1/2}) for all non-critical temperatures. In this paper we show that the mixing time is Theta(1) in high temperatures, Theta(log n) in low temperatures and Theta(n^{1/4}) at criticality. We also provide an upper bound of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on any tree with n vertices.
We consider a class of spin systems on $\Z^d$ with vector valued spins $(\bS_x)$ that interact via the pair-potentials $J_{x,y} \bS_x\cdot\bS_y$. The interactions are generally spread-out in the sense that the $J_{x,y}$'s exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions $d\ge3$, we can finally provide examples of the 3-state …
