Discontinuity of the phase transition for the planar random-cluster and Potts models with $q&gt;4$

Duminil-Copin, Hugo; Gagnebin, Maxime; Harel, Matan; Manolescu, Ioan; Tassion, Vincent

doi:10.48550/arxiv.1611.09877

Cited by 40 publications

(70 citation statements)

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“…Moreover, it was proved in [20] that φ 0 pc(q),q = φ 1 pc(q),q when 1 ≤ q ≤ 4. Conversely, when q > 4 it was proved in [15] (see also [41] for a shorter proof) that φ 0 pc(q),q ≠ φ 1 pc(q),q .…”

Section: Fk-percolationmentioning

confidence: 99%

“…The value T c (q) for which the phase transition occurs was determined in [2]. At T = T c (q), when 2 ≤ q ≤ 4 the Gibbs measure is unique [20] (continuous phase transition); when q > 4 the phase transition is discontinuous [15,41], and there exist at least q + 1 distinct Gibbs measures: one free measure obtained as a limit as T ↘ T c (q) and q monochromatic measures obtained as limits as T ↗ T c (q). The goal of the present paper is to prove that in this last case, the above-mentioned Gibbs measures are the only extremal ones.…”

Section: Introductionmentioning

confidence: 99%

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Structure of Gibbs measures for planar FK-percolation and Potts models

Glazman,

Manolescu

2021

Preprint

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We prove that all Gibbs measures of the q-state Potts model on Z 2 are linear combinations of the extremal measures obtained as thermodynamic limits under free or monochromatic boundary conditions. In particular all Gibbs measures are invariant under translations. This statement is new at points of first-order phase transition, that is at T = T c (q) when q > 4. In this case the structure of Gibbs measures is the most complex in the sense that there exist q + 1 distinct extremal measures.Most of the work is devoted to the FK-percolation model on Z 2 with q ≥ 1, where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are non-quantitative and follow the spirit of the seminal works of Aizenman [1] and Higuchi [32], which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation (i.e., a weaker spatial Markov property) pose serious additional difficulties compared to the case of the Ising model. For example, it is not automatic, albeit true, that thermodynamic limits are Gibbs. The result for the Potts model is then derived using the Edwards-Sokal coupling and auto-duality. The latter ingredient is necessary since applying the Edwards-Sokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FK-percolation.Finally, the proof is generic enough to adapt to the FK-percolation and Potts models on the triangular and hexagonal lattices and to the loop O(n) model in the range of parameters for which its spin representation is positively associated.

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Section: Fk-percolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Structure of Gibbs measures for planar FK-percolation and Potts models

Glazman,

Manolescu

2021

Preprint

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“…It is known [2] that the inverse critical temperature of a homogeneous system with Ising-like spins is equal to 2. This can be used to obtain the behavior of β c (p, q) near p = 1 by expanding (23) up to o(β c − 2), solving for β c and then taking the leading order term in 1 − p, to give…”

Section: Mean Field Modelmentioning

confidence: 99%

“…We shall now provide a heuristic argument to supports the existence of p * for the MF model. First, note that p = 0, β c = q simultaneously solve (23). Then, assuming that the phase portrait is continuous, in particular at p = 0, we have that it must converge to q in the limit p → 0.…”

Section: Mean Field Modelmentioning

confidence: 99%

“…Based on these findings, Baxter conjectured that the model in subject exhibits a continuous transition for q ≤ q c and discontinuous transition for q > q c with q c = 4 being the changeover integer or the maximal integer for which the transition is continuous. Recently, Duminil-Copin et al [22,23] have rigorously proven Baxter's conjecture using the random cluster or the Fortuin-Kasteleyn (FK) [24] representation of the Potts model. Though the changeover integer q c = 4 has been found for the square lattice, it is believed to be lattice independent [7].…”

Section: Introductionmentioning

confidence: 99%

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Changeover phenomenon in randomly colored Potts models

Schreiber,

Cohen,

Amir

et al. 2021

Preprint

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A hybrid Potts model where a random concentration p of the spins assume q0 states and a random concentration 1−p of the spins assume q > q0 states is introduced. It is known that when the system is homogeneous, with an integer spin number q0 or q, it undergoes a second or a first order transition, respectively. It is argued that there is a concentration p * such that the transition nature of the model is changed at p * . This idea is demonstrated analytically and by simulations for two different types of interaction: the usual square lattice nearest neighboring and the mean field all-to-all interaction. Exact expressions for the second order critical line in concentration-temperature parameter space of the mean field model together with some other related critical properties, are derived.

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Finitary codings for gradient models and a new graphical representation for the six‐vertex model

Ray

Spinka

2021

Random Struct Algorithms

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It is known that the Ising model on Z 𝑑 at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction to being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume-order large deviation estimate for the energy. Results in the same spirit are shown for the Potts model, the so-called beach model, and the six-vertex model. We also introduce a coupling between the six-vertex model with c ≥ 2 and a new Edwards-Sokal type graphical representation of it, which we believe is of independent interest.

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Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

Cited by 40 publications

References 0 publications

Structure of Gibbs measures for planar FK-percolation and Potts models

Structure of Gibbs measures for planar FK-percolation and Potts models

Changeover phenomenon in randomly colored Potts models

Finitary codings for gradient models and a new graphical representation for the six‐vertex model

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