Ising models on locally tree-like graphs
Abstract
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.
Cite
@article{arxiv.0804.4726,
title = {Ising models on locally tree-like graphs},
author = {Amir Dembo and Andrea Montanari},
journal= {arXiv preprint arXiv:0804.4726},
year = {2016}
}
Comments
Published in at http://dx.doi.org/10.1214/09-AAP627 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)