Ising models on locally tree-like graphs

Details Ising models on locally tree-like graphs Ising models on locally tree-like graphs

2008-04-30

arxiv.org/abs/0804.4726v3

Annals of Applied Probability 2010, Vol. 20, No. 2, 565-592

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AAP627 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

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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