Potts and random cluster measures on locally regular-tree-like graphs
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| Format: | Preprint |
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2023
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| _version_ | 1866913849094963200 |
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| author | Basak, Anirban Dembo, Amir Sly, Allan |
| author_facet | Basak, Anirban Dembo, Amir Sly, Allan |
| contents | 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 $φ^{q,β,B}_{n}$. 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 $φ^{q,β,B}_n$ 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 $k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_16008 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Potts and random cluster measures on locally regular-tree-like graphs Basak, Anirban Dembo, Amir Sly, Allan Probability Statistical Mechanics Mathematical Physics 60K35, 82B20, 82B26 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 $φ^{q,β,B}_{n}$. 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 $φ^{q,β,B}_n$ 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 $k$. |
| title | Potts and random cluster measures on locally regular-tree-like graphs |
| topic | Probability Statistical Mechanics Mathematical Physics 60K35, 82B20, 82B26 |
| url | https://arxiv.org/abs/2312.16008 |