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

Amir Dembo, Andrea Montanari, Allan Sly, Nike Sun

Research output: Contribution to journalArticlepeer-review

43 Scopus citations


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 (US)
Pages (from-to)551-575
Number of pages25
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Apr 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


Dive into the research topics of 'The Replica Symmetric Solution for Potts Models on d-Regular Graphs'. Together they form a unique fingerprint.

Cite this