Random Cluster Model on Regular Graphs

Bencs, Ferenc and Borbényi, Márton and Csíkvári, Péter (2023) Random Cluster Model on Regular Graphs. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 399. pp. 203-248. ISSN 0010-3616

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Abstract

For a graph G=(V,E) G = ( V , E ) with v ( G ) vertices the partition function of the random cluster model is defined by \begin{aligned} Z_G(q,w)=\sum _{A\subseteq E(G)}q^{k(A)}w^{|A|}, \end{aligned} Z G ( q , w ) = ∑ A ⊆ E ( G ) q k ( A ) w | A | , where k ( A ) denotes the number of connected components of the graph ( V , A ). Furthermore, let g ( G ) denote the girth of the graph G , that is, the length of the shortest cycle. In this paper we show that if (G_n)_n ( G n ) n is a sequence of d -regular graphs such that the girth g(G_n)\rightarrow \infty g ( G n ) → ∞ , then the limit \begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{v(G_n)}\ln Z_{G_n}(q,w)=\ln \Phi _{d,q,w} \end{aligned} lim n → ∞ 1 v ( G n ) ln Z G n ( q , w ) = ln Φ d , q , w exists if q\ge 2 q ≥ 2 and w\ge 0 w ≥ 0 . The quantity \Phi _{d,q,w} Φ d , q , w can be computed as follows. Let \begin{aligned} \Phi _{d,q,w}(t):= & {} \left( \sqrt{1+\frac{w}{q}}\cos (t)+\sqrt{\frac{(q-1)w}{q}}\sin (t)\right) ^{d}\{} & {} +\,(q-1)\left( \sqrt{1+\frac{w}{q}}\cos (t) -\sqrt{\frac{w}{q(q-1)}}\sin (t)\right) ^{d}, \end{aligned} Φ d , q , w ( t ) : = 1 + w q cos ( t ) + ( q - 1 ) w q sin ( t ) d + ( q - 1 ) 1 + w q cos ( t ) - w q ( q - 1 ) sin ( t ) d , then \begin{aligned} \Phi _{d,q,w}:=\max _{t\in [-\pi ,\pi ]}\Phi _{d,q,w}(t), \end{aligned} Φ d , q , w : = max t ∈ [ - π , π ] Φ d , q , w ( t ) , The same conclusion holds true for a sequence of random d -regular graphs with probability one. Our result extends the work of Dembo, Montanari, Sly and Sun for the Potts model (integer q ), and we prove a conjecture of Helmuth, Jenssen and Perkins about the phase transition of the random cluster model with fixed q .

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