Potts and random cluster measures on locally regular-tree-like graphs

TL;DR

The paper rigorously links ferromagnetic Potts models and their random-cluster representations on large locally tree-like graphs to Bethe-approximate free energies. By using an Edwards–Sokal coupling and a detailed analysis of Bethe recursions, it proves that local weak limits of Potts and rc measures converge to free or wired tree measures, with mixtures appearing on the critical line where Bethe energies coincide. A dominating-color pure-state decomposition is established for $B=0$ when $\beta>\beta_c(0)$ under uniform edge expansion, while in the disordered phase ($\beta<\beta_c(0)$) all colors coalesce to the same limit. The methodology hinges on rcms and their monotonicity, tail-trivial messages, and a careful comparison of free-energy densities, providing rigorous backing for the Bethe replica-symmetric predictions in this ferromagnetic Potts setting. The results illuminate the interplay between boundary conditions, phase structure, and local weak limits on locally tree-like graphs, with implications for mean-field behavior and algorithmic/metastability questions.

Abstract

Fixing $β\ge 0$ and an integer $q \ge 2$, consider the ferromagnetic $q$-Potts measures $μ_n^{β,B}$ on finite graphs ${\sf G}_n$ on $n$ vertices, with external field strength $B \ge 0$ and the corresponding random cluster measures $\varphi^{q,β,B}_{n}$. Suppose that as $n \to \infty$ the uniformly sparse graphs ${\sf G}_n$ converge locally to an infinite $d$-regular tree ${\sf T}_{d}$, $d \ge 3$. We show that the convergence of the Potts free energy density to its Bethe replica symmetric prediction (which has been proved in case $d$ is even, or when $B=0$), yields the local weak convergence of $\varphi^{q,β,B}_n$ and $μ_n^{β,B}$ to the corresponding free or wired random cluster measure, Potts measure, respectively, on ${\sf T}_{d}$. The choice of free versus wired limit is according to which has the larger Potts Bethe functional value, with mixtures of these two appearing {as limit points on} the critical line $β_c(q,B)$ where these two values of the Bethe functional coincide. For $B=0$ and $β>β_c$, we further establish a pure-state decomposition by showing that conditionally on the same dominant color $1 \le k \le q$, the $q$-Potts measures on such edge-expander graphs ${\sf G}_n$ converge locally to the $q$-Potts measure on ${\sf T}_{d}$ with a boundary wired at color $k$.

Figures (1)

• Figure 1: The left panel shows the non-uniqueness regime for the Potts measures on ${\sf T}_d$ for $q=30$ and $d=3$ (see Proposition \ref{['prop:non-unique-regime']} for its definition). The lightly shaded region is $\sf R_{\mathop{\mathrm{f}}\nolimits}$, while the darker region is $\sf R_1$. These two regions are separated by the critical curve $\beta_c(B)$, and the non-uniqueness regime $\sf R_{\neq}$ is separated from its complement by the two curves $\beta_{\mathop{\mathrm{f}}\nolimits}(B)$ and $\beta_+(B)$. The right panel shows the non-uniqueness regime along with the local weak limits for Potts measures for $q=30$ and $d=10$. Notice the difference in shapes and sizes of $\sf R_1$ and $\sf R_{\mathop{\mathrm{f}}\nolimits}$ compared to the left panel. Note that Theorems \ref{['thm:main-1']} and \ref{['thm:crit-1']} identify the local weak limit (points) except when $(\beta,B)=(\beta_c(0),0)$, marked black in the right panel.

Theorems & Definitions (73)

• Definition 1.1: Uniform sparsity and local weak convergence of graphs
• Definition 1.2: Potts on trees, with given boundary conditions
• Definition 1.3: Bethe functional; Bethe recursion and its fixed points
• Proposition 1.4
• Proposition 1.5: Description of the non-uniqueness region
• Definition 1.6: Amending graphs by a ghost vertex
• Definition 1.7: Random cluster measure for a finite graph
• Remark 1.8
• Definition 1.9: The free and the wired rcm-s on ${\sf T}_d$
• Definition 1.10: Spaces of probability measures