Gibbsianness and non-Gibbsianness in divide and color models

Bálint, András

doi:10.1214/09-aop518

Cited by 10 publications

(25 citation statements)

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“…Interesting sources of non-Gibbsian behavior include time evolutions or deterministic transformations which reduce the complexity of the local state space. A prototypical example of a system of the second type is the fuzzy Potts model (fuzzy PM) [20,28,26,22,19,1]. It is obtained from the ordinary PM by partitioning the local state space {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction

Jahnel

Külske²

2017

Bernoulli

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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 Fernández, den Hollander and Martínez [18] 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 use monotocity arguments.

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Section: Introductionmentioning

confidence: 99%

Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction

Jahnel

Külske²

2017

Bernoulli

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“…Since p c (2) = 1 − e −βc , β < β c implies p < p c (2), so that ν p,2 is well-defined, and the FK clusters are all finite with probability one.…”

Section: Resultsmentioning

confidence: 99%

“…They both follow the structure of Russo's formulation [30] of Kesten's proof of p c = 1/2 for independent bond percolation on Z 2 , but in both cases, sophisticated methods are required to deal with issues of dependence. Higuchi's proof makes heavy use of the Markovianness of the Ising model for all values of h, but in our model, this property holds only at r = 1/2, and fails for all other values of r (see [2]). Therefore we instead utilise the fact that at r = 1/2 our model coincides with the Ising model, so that we can use a lemma by Higuchi, together with new domination lemmas (presented in Sect.…”

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confidence: 87%

“…1. Let E + be the event that there exists an infinite (+)-cluster somewhere in T. By the ergodicity of P β,r for β < β c (which follows from the ergodicity of ν p,2 for all p < p c (2), see [16]), we have that P β,r (E + ) is 1 for all r > 1/2 and 0 for all r < 1/2. Now, for a bond configuration η ∈ {0, 1} E T , let G η denote the realisation of the random graph corresponding to η as defined in Sect.…”

Section: Theorem 1 For Allmentioning

confidence: 99%

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The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

2010

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We define a new percolation model by generalising the FK representation of the Ising model, and show that on the triangular lattice and at high temperatures, the critical point in the new model corresponds to the Ising model. Since the new model can be viewed as Bernoulli percolation on a random graph, our result makes an explicit connection between Ising percolation and critical Bernoulli percolation, and gives a new justification of the conjecture that the high temperature Ising model on the triangular lattice is in the same universality class as Bernoulli percolation.Keywords Ising model · Random-cluster measures · Dependent percolation · DaC models · Sharp phase transition · Duality · p c = 1/2 Motivation, Background and SynopsisIf one considers the percolation properties of spin clusters, the high temperature (β < β c ) Ising model on the triangular lattice T with no external field shows critical behaviour in the sense that the mean spin cluster size is divergent (see, e.g., [3]) and the probability that two spins are in the same spin cluster has a power law decay (see [17]). The relevant universality class is conjectured to be that of two-dimensional Bernoulli (independent) percolation. This conjecture includes convergence of certain interfaces to SLE 6 in the scaling limit, and

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“…Again, we only need to look at n = 2 and 3. These are however contained in (A) since (i) it is easier to be injective on a subset (in fact, in this case, RER [2] = RER exch [2] ) and (ii) the examples there showing non-injectivity for n = 3 are in fact exchangeable.…”

Section: Uniqueness Of the Representing Rer In The Finite Casementioning

confidence: 99%

Generalized Divide and Color Models

Steif¹,

Tykesson²

2019

ALEA

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In this paper, we initiate the study of "Generalized Divide and Color Models". A very special interesting case of this is the "Divide and Color Model" (which motivates the name we use) introduced and studied by Olle Häggström.In this generalized model, one starts with a finite or countable set V , a random partition of V and a parameter p ∈ [0, 1]. The corresponding Generalized Divide and Color Model is the {0, 1}-valued process indexed by V obtained by independently, for each partition element in the random partition chosen, with probability p, assigning all the elements of the partition element the value 1, and with probability 1 − p, assigning all the elements of the partition element the value 0.Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have?The motivation for studying these processes is twofold; on the one hand, we believe that this is a very natural and interesting class of processes that deserves investigation and on the other hand, a number of quite varied well-studied processes actually fall into this class such as (1) the Ising model, (2) the fuzzy Potts model, (3) the stationary distributions for the Voter Model, (4) random walk in random scenery and of course (5) the original Divide and Color Model. Contents 1 Introduction 2 2 The finite case 8 3 Color processes associated to infinite exchangeable random partitions 18 4 Connected random equivalence relations on Z 33 5 Stochastic domination of product measures 38 6 Ergodic results in the translation invariant case 46 7 Questions and further directions 56 Definition 1.1. The Ising model onIt turns out that µ G,J,0 is a color process when J ≥ 0; this corresponds to the famous FK (Fortuin-Kasteleyn) or so-called random cluster representation. To explain this, we first need to introduce the following model.where N 1 is the number of edges in state 1, N 2 is the number of edges in state 0, C is the resulting number of connected clusters and Z = Z(G, α, q) is a normalization constant.Note, if q = 1, this is simply an i.i.d. process with parameter α. We think of ν RCM G,α,q as an RER on V by looking at the clusters of the percolation realization; i.e., v and w are in the same partition if there is a path from v to w using edges in state 1.The following theorem from [14] tells us that the Ising Model with J ≥ 0 and h = 0 is indeed a color process. We however must identify −1 with 0. See also [11]. Theorem 1.3. ([11],[14]) For any graph G and any J ≥ 0, µ G,J,0 = Φ 1/2 (ν RCM G,1−e −2J ,2 ).See [22] for a nice survey concerning various random cluster representations. We remark that while for all p, Φ p (ν RCM G,α,2 ) is of course a color process, we do not know if this corresponds to anything natural wh...

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Gibbsianness and non-Gibbsianness in divide and color models

Cited by 10 publications

References 29 publications

Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction

Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

Generalized Divide and Color Models

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