Random Cluster Model on Regular Graphs

Abstract

AbstractFor 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.