Potts and Random Cluster Measures on Locally Regular-Tree-Like Graphs

Fixing beta >= 0 and an integer q >= 2 consider the ferromagnetic q-Potts B.B measures mu(beta,B)(n) on finite graphs G, on n vertices, with external field strength B >= 0 .B.B and the corresponding random cluster measures Phi(q beta.B)(n) Suppose that as n the uniformly sparse graphs G, converge locally to an infinite d-regular tree T-d, d >= 3 We show that the convergence of the Potts free energy density to its Bethe-Peirles replica symmetric prediction (which has been proved in case d is even, or when B = 0 ) yields the local weak convergence of Phi(q beta.B)(n) and mu(beta,B)(n) to the corresponding free or wired random cluster measure, Potts measure, respectively, on T-d. The choice of free versus wired limit is according to which has the larger Potts Bethe functional value, with mixtures of these two appearing as limit points on the critical line beta(c)(q, B) where these two values of the Bethe functional coincide. For B = 0 and beta > B-c. we further establish a pure-state decomposition by showing that conditionally on the same dominant color 1 <= k <= q . the q-Potts measures on such edge-expander graphs Ga converge locally to the q-Potts measure on T-d with a boundary wired at color k.