Factor models on locally tree-like graphs

TL;DR

This paper rigorously analyzes factor models on sparse graphs converging to trees, establishing the existence and explicit computation of the free energy density using a new interpolation scheme, and connecting it to Bethe free energy and physics heuristics.

Contribution

It introduces a novel interpolation method to compute the free energy density for factor models on locally tree-like graphs, extending Bethe approximation rigorously beyond physics heuristics.

Findings

Explicit formulas for free energy in various models

Bounds on free energy beyond uniqueness regimes

Connection of free energy to Bethe fixed points

Abstract

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $\phi$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $\phi$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $\phi$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on $\phi$. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on $T$. In the special case that $T$ has a Galton-Watson law, this formula coincides with the nonrigorous "Bethe prediction" obtained by statistical physicists using the "replica" or "cavity" methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.