FACTOR MODELS ON LOCALLY TREE-LIKE GRAPHS

• 成果类型: Article
• 署名作者: Dembo, Amir; Montanari, Andrea; Sun, Nike
• 署名单位: Stanford University; Stanford University; Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP828
• 发表日期: 2013
• 页码: 4162-4213
• 关键词: markov random-fields; independent sets; replica bounds; optimization; hardness

摘要

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.