Probability and Mathematical Physics Seminar

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Fixing integers q,d>2, denote by Q(n,T,B) the ferromagnetic q-Potts measures on graphs G(n), at temperature T>0 and non-negative external field strength B, where as n grows the uniformly sparse G(n) of n vertices converge locally to the infinite d-regular tree. I will review a joint work with Anirban Basak and Allan Sly, showing that the convergence of the Potts free energy density to its Bethe replica symmetric prediction (which was proved for even d, or when B=0), yields the local weak convergence of Q(n,T,B) to the corresponding free or wired Potts measure on the infinite tree. One gets the free versus wired limit, according to which has the larger Potts Bethe functional value, with mixtures of these two appearing as limit points at the critical temperature T_c(q,B), where these two values of the Bethe functional coincide. For edge-expander G(n), we also establish a pure-state decomposition by showing that below the critical temperature, conditionally on having a dominant color k, the measures Q(n,T,0) converge locally to the q-Potts measure on the infinite tree, with a boundary wired at color k.