Factor models on locally treelike 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 $\phi$, the limit of the logpartition 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 hardcore) 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 GaltonWatson 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 DOI:
 10.48550/arXiv.1110.4821
 arXiv:
 arXiv:1110.4821
 Bibcode:
 2011arXiv1110.4821D
 Keywords:

 Mathematics  Probability;
 Computer Science  Discrete Mathematics
 EPrint:
 Published in at http://dx.doi.org/10.1214/12AOP828 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)