Spin systems on Bethe lattices

TL;DR

This paper rigorously proves the decomposition of spin systems on Bethe lattices into Bethe states and provides a method to compute the free energy, confirming and extending the cavity method's predictions.

Contribution

It establishes the decomposition into Bethe states and a variational formula for free energy for general spin systems on Bethe lattices.

Findings

Rigorous proof of Bethe state decomposition

Derivation of a variational formula for free energy

Validation of the cavity method's assumptions

Abstract

In an extremely influential paper Mezard and Parisi put forward an analytic but non-rigorous approach called the cavity method for studying spin systems on the Bethe lattice, i.e., the random $d$-regular graph [Eur. Phys. J. B 20 (2001) 217--233]. Their technique was based on certain hypotheses; most importantly, that the phase space decomposes into a number of Bethe states that are free from long-range correlations and whose marginals are given by a recurrence called Belief Propagation. In this paper we establish this decomposition rigorously for a very general family of spin systems. In addition, we show that the free energy can be computed from this decomposition. We also derive a variational formula for the free energy. The general results have interesting ramifications on several special cases.