Bethe states of random factor graphs

TL;DR

This paper confirms a key aspect of the replica symmetry breaking hypothesis for random factor graph models, showing that the Gibbs measure decomposes into Bethe states with simple correlation structures, linked to Belief Propagation fixed points.

Contribution

It proves that the Gibbs measure can be decomposed into Bethe states for a broad class of models, connecting these states to Belief Propagation fixed points and introducing a new approximation result.

Findings

Gibbs measure decomposes into a moderate number of Bethe states.

Bethe states' marginals are derived from Belief Propagation fixed points.

New approximation method for probability measures on discrete cubes.

Abstract

We verify a key component of the replica symmetry breaking hypothesis put forward in the physics literature [M\'ezard and Montanari 2009] on random factor graph models. For a broad class of these models we verify that the Gibbs measure can be decomposed into a moderate number of Bethe states, subsets of the state space in which both short and long range correlations of the measure take a simple form. Moreover, we show that the marginals of these Bethe states can be obtained from fixed points of the Belief Propagation operator. We derive these results from a new result on the approximation of general probability measures on discrete cubes by convex combinations of product measures.