### 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 language | English (US) |
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Pages (from-to) | 551-575 |

Number of pages | 25 |

Journal | Communications In Mathematical Physics |

Volume | 327 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2014 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*327*(2), 551-575. https://doi.org/10.1007/s00220-014-1956-6