Open Access
April 2010 Ising models on locally tree-like graphs
Amir Dembo, Andrea Montanari
Ann. Appl. Probab. 20(2): 565-592 (April 2010). DOI: 10.1214/09-AAP627


We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the “cavity” prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.


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Amir Dembo. Andrea Montanari. "Ising models on locally tree-like graphs." Ann. Appl. Probab. 20 (2) 565 - 592, April 2010.


Published: April 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1191.82025
MathSciNet: MR2650042
Digital Object Identifier: 10.1214/09-AAP627

Primary: 82B44
Secondary: 05C05 , 05C80 , 60F10 , 60K35 , 82B23

Keywords: belief propagation , Bethe measures , Cavity method , Ising model , Local weak convergence , random sparse graphs

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.20 • No. 2 • April 2010
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