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arXiv · 1808.03440

Spin systems on Bethe lattices

Amin Coja-Oghlan·Will Perkins
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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.

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Amin Coja-Oghlan, Will Perkins. 2018-08-10. Spin systems on Bethe lattices. https://doi.org/10.1007/s00220-019-03544-y

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