Random Cluster Model on Regular Graphs

For a graph 𝐺=(𝑉,𝐸) with v(G) vertices the partition function of the random cluster model is defined by

ZG (q,w) = Σ qk(A)ω|A|
A⊂E(G)

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 (𝐺𝑛)𝑛 is a sequence of d-regular graphs such that the girth 𝑔(𝐺𝑛)→∞, then the limit

lim _n→∞ ¹ _v(Gn) ln Z_Gn (q,w)= ln ᶲ d,q,w

exists if q ≥ 2 and w ≥ 0. The quantity d,q,w can be computed as follows. Let

ᶲ _d,q,w(t) := (√1+^w/q cos(t) + √(q −1)w/_q sin(t) ^d

+(q −1) (√ 1+^w/q cos(t) − √d,q,w := max w/_{q(q −1)} sin(t)^d

then

ᶲd,q,w : = max ᶲ_d,q,w(^t),
                 tε[-ππ]

The same conclusion holds true for a sequence of random d-regular graphs with probalitity one. Our result extends the work of Dembo, Montanari, Sly and Sun for the Potts model (integer q), and we prove a conjecture of Helumth, Jenssen and Perskins about the phase transition of the random cluster model with fixed q.