Potts and random cluster measures on locally regular-tree-like graphs

Fixing β≥ 0 and an integer q ≥ 2, consider the ferromagnetic q-Potts measures μ_n^β,B on finite graphs G_n on n vertices, with external field strength B ≥ 0 and the corresponding random cluster measures φ^q,β,B_n. Suppose that as n →∞ the uniformly sparse graphs G_n 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 replica symmetric prediction (which has been proved in case d is even, or when B=0), yields the local weak convergence of φ^q,β,B_n and μ_n^β,B 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 β_c(q,B) where these two values of the Bethe functional coincide. For B=0 and β>β_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 G_n converge locally to the q-Potts measure on T_d with a boundary wired at color k.