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Information Center for Mathematical Science

PAC

THE REPLICA SYMMETRIC SOLUTION FOR POTTS MODELS ON d-REGULAR GRAPHS
Author AMIR DEMBO (Stanford University)
Homepage Url http://www-stat.stanford.edu/~adembo/
Coauthors ANDREA MONTANARI, ALLAN SLY, NIKE SUN
Abstract We provide an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to the d-regular tree for d even, covering all temperature regimes. This formula coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d-regular tree, the so-called replica symmetric solution. For uniformly random d-regular graphs we further show that the replica symmetric Bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions.
Abstract Url http://www-stat.stanford.edu/~adembo/preprints/d-regular.pdf