The Replica Symmetric Solution for Potts Models on d-Regular Graphs

Amir Dembo, Andrea Montanari, Allan Sly, Nike Sun*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    43 Citations (Scopus)

    Abstract

    We establish 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.

    Original languageEnglish
    Pages (from-to)551-575
    Number of pages25
    JournalCommunications in Mathematical Physics
    Volume327
    Issue number2
    DOIs
    Publication statusPublished - Apr 2014

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