Open Access
November 2013 Factor models on locally tree-like graphs
Amir Dembo, Andrea Montanari, Nike Sun
Ann. Probab. 41(6): 4162-4213 (November 2013). DOI: 10.1214/12-AOP828


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.


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Amir Dembo. Andrea Montanari. Nike Sun. "Factor models on locally tree-like graphs." Ann. Probab. 41 (6) 4162 - 4213, November 2013.


Published: November 2013
First available in Project Euclid: 20 November 2013

zbMATH: 1280.05119
MathSciNet: MR3161472
Digital Object Identifier: 10.1214/12-AOP828

Primary: 05C80 , 60K35 , 82B20 , 82B23

Keywords: belief propagation , Bethe measures , factor models , free energy density , Gibbs measures , Independent set , Local weak convergence , Potts model , Random graphs

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 6 • November 2013
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