The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1

Abstract

We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self-dual pointThis gives a proof that the critical temperature of the q-state Potts model is equal to log(1 + √ q) for all q 2. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all q 1, in contrast to earlier methods valid only for certain given q. The proof extends to the triangular and the hexagonal lattices as well. Mathematics Subject Classification (2000)60K35 · 82B20 (primary); 82B26 · 82B43

The conformal loop ensemble nesting field

“…For the index i = 1, we write w) 2 , where G U denotes the Green's function for the Laplacian in the domain U . By the Koebe distortion theorem, the Green's function is in turn bounded by G U (z, w) ≤ const + const × max(0, log(CR(z; U )/|z − w|)).…”

“…Then we sample the negative log conformal radii of the loops of 1 and 2 surrounding 0, so as to maximize the probability that these coincide with λ (0) on − 3 4 x, ∞ . If either λ (1) or λ (2) does not coincide with λ (0) on − a , then we generate the restriction of i to these components independently of everything else generated thus far. The resulting loop processes i are distributed according to the conformal loop ensemble on D i , and have been coupled to be similar near 0.…”

“…Let ψ = ψ (2) • (ψ (1) ) −1 be the conformal map from U (1) a to U (2) a for which ψ(0) = 0 and ψ (0) = 1. Assuming a ∈ (− 3 4 x, − 1 2 x) and a ∈ λ (1) and a ∈ λ (2) , the Koebe distortion theorem implies that on B(0, e x/4 ), |ψ − 1| is exponentially small in x.…”

“…Assuming a ∈ (− 3 4 x, − 1 2 x) and a ∈ λ (1) and a ∈ λ (2) , the Koebe distortion theorem implies that on B(0, e x/4 ), |ψ − 1| is exponentially small in x.…”

“…2 Nesting of loops separating critical Fortuin-Kasteleyn (FK) clusters from dual clusters. Each FK bond configuration has probability proportional to ( p/(1 − p)) # edges × q # clusters [4], where there is believed to be a critical point at p = 1/(1 + 1/ √ q) (proved for q ≥ 1 [2]). For 0 ≤ q ≤ 4, these loops are believed to have the same large-scale behavior as the O(n) model loops for n = √ q in the dense phase, that is, to converge to CLE κ for 4 ≤ κ ≤ 8 (see [6,11]), a Critical FK bond configuration.…”