The Annals of Probability

Factor models on locally tree-like graphs

Amir Dembo, Andrea Montanari, and Nike Sun

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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.

Article information

Ann. Probab., Volume 41, Number 6 (2013), 4162-4213.

First available in Project Euclid: 20 November 2013

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B23: Exactly solvable models; Bethe ansatz

Factor models random graphs belief propagation Bethe measures Potts model independent set Gibbs measures free energy density local weak convergence


Dembo, Amir; Montanari, Andrea; Sun, Nike. Factor models on locally tree-like graphs. Ann. Probab. 41 (2013), no. 6, 4162--4213. doi:10.1214/12-AOP828.

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