Characterizing the limiting critical Potts measures on locally regular-tree-like expander graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866913866919706624 |
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| author | Du, Hang Zhou, Yanxin |
| author_facet | Du, Hang Zhou, Yanxin |
| contents | For any integers $d,q\ge 3$, we consider the $q$-state ferromagnetic Potts model with an external field on a sequence of expander graphs that converges to the $d$-regular tree $\mathtt{T}_d$ in the Benjamini-Schramm sense. We show that along the critical line, any subsequential local weak limit of the Potts measures is a mixture of the free and wired Potts Gibbs measures on $\mathtt{T}_d$. Furthermore, we show the possibility of an arbitrary extent of strong phase coexistence: for any $α\in [0,1]$, there exists a sequence of locally $\mathtt{T}_d$-like expander graphs $\{G_n\}$, such that the Potts measures on $\{G_n\}$ locally weakly converges to the $(α,1-α)$-mixture of the free and wired Potts Gibbs measures. Our result extends results of \cite{HJP23} which restrict to the zero-field case and also require $q$ to be sufficiently large relative to $d$, and results of \cite{BDS23} which restrict to the even $d$ case. We also confirm the phase coexistence prediction of \cite{BDS23}, asserting that the Potts local weak limit is a genuine mixture of the free and wired states in a generic setting. We further characterize the subsequential local weak limits of random cluster measures on such graph sequences, for any cluster parameter $q>2$ (not necessarily integer). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing the limiting critical Potts measures on locally regular-tree-like expander graphs Du, Hang Zhou, Yanxin Probability Mathematical Physics 60K35, 82B20, 82B27 For any integers $d,q\ge 3$, we consider the $q$-state ferromagnetic Potts model with an external field on a sequence of expander graphs that converges to the $d$-regular tree $\mathtt{T}_d$ in the Benjamini-Schramm sense. We show that along the critical line, any subsequential local weak limit of the Potts measures is a mixture of the free and wired Potts Gibbs measures on $\mathtt{T}_d$. Furthermore, we show the possibility of an arbitrary extent of strong phase coexistence: for any $α\in [0,1]$, there exists a sequence of locally $\mathtt{T}_d$-like expander graphs $\{G_n\}$, such that the Potts measures on $\{G_n\}$ locally weakly converges to the $(α,1-α)$-mixture of the free and wired Potts Gibbs measures. Our result extends results of \cite{HJP23} which restrict to the zero-field case and also require $q$ to be sufficiently large relative to $d$, and results of \cite{BDS23} which restrict to the even $d$ case. We also confirm the phase coexistence prediction of \cite{BDS23}, asserting that the Potts local weak limit is a genuine mixture of the free and wired states in a generic setting. We further characterize the subsequential local weak limits of random cluster measures on such graph sequences, for any cluster parameter $q>2$ (not necessarily integer). |
| title | Characterizing the limiting critical Potts measures on locally regular-tree-like expander graphs |
| topic | Probability Mathematical Physics 60K35, 82B20, 82B27 |
| url | https://arxiv.org/abs/2505.24283 |