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 language||English (US)|
|Number of pages||25|
|Journal||Communications In Mathematical Physics|
|State||Published - Apr 2014|
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics