The replica symmetric solution for Potts models on d-regular graphs

2012/07/23 by Amir Dembo, Andrea Montanari, Dembo, Amir +5
Mathematics · Physics and Astronomy · #05C80 #60K35 #82B20 #82B23 #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech #math.PR #msc:05C80 #msc:60K35 #msc:82B20 #msc:82B23

paper · pdf · doi:10.48550/arxiv.1207.5500

23 pages

arxiv created 2012/07/23 · arxiv updated 2012/07/24

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.

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