Random cluster model on regular graphs

Abstract

For a graph $G=(V,E)$ with $v(G)$ vertices the partition function of the random cluster model is defined by $$Z_G(q,w)=\sum _{A\subseteq E(G)}{q}^{k(A)}{w}^{|A|},$$ 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 $(G_n{)}_n$ is a sequence of $d$-regular graphs such that the girth $g(G_n)\to \infty$, then the limit $$\underset{n\to \infty }{\lim }\frac{1}{v(G_n)}\ln Z_{G_n}(q,w)=\ln {\Phi }_{d,q,w}$$ exists if $q\ge 2$ and $w\ge 0$. The quantity ${\Phi }_{d,q,w}$ can be computed as follows. Let $${\Phi }_{d,q,w}(t):={\left(\sqrt{1+\frac{w}{q}}\cos (t)+\sqrt{\frac{(q-1)w}{q}}\sin (t)\right)}^d+(q-1){\left(\sqrt{1+\frac{w}{q}}\cos (t)-\sqrt{\frac{w}{q(q-1)}}\sin (t)\right)}^d,$$ then $${\Phi }_{d,q,w}:=\underset{t\in [-\pi ,\pi ]}{\max }{\Phi }_{d,q,w}(t),$$ 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 $q$.

38 pages

References in corpus (1)

• Loop series for discrete statistical models on graphs

Cited by in corpus (1)

• Sampling from the random cluster model on random regular graphs at all temperatures via Glauber dynamics