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 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). Further- more, 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 ) → ∞, then the limit lim n→∞ 1 v(G n ) ln Z G n (q, w) = ln d,q,w exists if q ≥ 2 and w ≥ 0. The quantity d,q,w can be computed as follows. Let 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 d,q,w := max t∈[−π,π] d,q,w (t), The same conclusion holds true for a sequence of random d-regular graphs with proba- bility 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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