Factor models on locally tree-like graphs

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 $\varphi$, 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 $\varphi$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $\varphi$ 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 $\varphi$. 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.

Published in at http://dx.doi.org/10.1214/12-AOP828 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models
• Ising models on power-law random graphs

Cited by in corpus (12)

• Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models
• Information-theoretic thresholds from the cavity method
• Evolutionary potential games on lattices
• Limits of discrete distributions and Gibbs measures on random graphs
• Spin systems on Bethe lattices
• The random 2-SAT partition function
• Metastability of the Potts ferromagnet on random regular graphs
• The set of solutions of random XORSAT formulae
• Random-cluster dynamics on random regular graphs in tree uniqueness
• Gibbs measures over locally tree-like graphs and percolative entropy over infinite regular trees
• Random cluster model on regular graphs
• Non-robust phase transitions in the generalized clock model on trees