Spin systems on Bethe lattices
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 -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.
Keywords
Cite
@article{arxiv.1808.03440,
title = {Spin systems on Bethe lattices},
author = {Amin Coja-Oghlan and Will Perkins},
journal= {arXiv preprint arXiv:1808.03440},
year = {2019}
}