Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Potts and random cluster measures on locally regular-tree-like graphs (2312.16008v3)

Published 26 Dec 2023 in math.PR, cond-mat.stat-mech, math-ph, and math.MP

Abstract: Fixing $\beta \ge 0$ and an integer $q \ge 2$, consider the ferromagnetic $q$-Potts measures $\mu_n{\beta,B}$ on finite graphs ${\sf G}n$ on $n$ vertices, with external field strength $B \ge 0$ and the corresponding random cluster measures $\varphi{q,\beta,B}{n}$. 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 $\varphi{q,\beta,B}_n$ and $\mu_n{\beta,B}$ 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 $\beta_c(q,B)$ where these two values of the Bethe functional coincide. For $B=0$ and $\beta>\beta_c$, 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 $k$.

Citations (1)

Summary

We haven't generated a summary for this paper yet.

X Twitter Logo Streamline Icon: https://streamlinehq.com