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 μnβ,B on finite graphs Gn on n vertices, with external field strength B≥0 and the corresponding random cluster measures φnq,β,B. Suppose that as n→∞ the uniformly sparse graphs Gn converge locally to an infinite d-regular tree Td, 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 φnq,β,B and μnβ,B to the corresponding free or wired random cluster measure, Potts measure, respectively, on Td. 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 βc(q,B) where these two values of the Bethe functional coincide. For B=0 and β>βc, 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 Gn converge locally to the q-Potts measure on Td with a boundary wired at color k.