Title : Belief propagation bounds for Ising phase transitions on graphs
Abstract : Unlike the mean field approximation which gives an upper bound of free energy, Belief propagation (BP, also known as Bethe-Peierls expansion) is commonly viewed as an ad-hock approximation, albeit potentially more accurate. I will argue that, nevertheless, BP can be used to bound the pair correlation function and per-spin-susceptibility in the ferromagnetic Ising model on an arbitrary graph, where spins are placed on graph vertices, and an interaction term corresponds to each graph edge. I will also present a related inequality for disorder-averaged susceptibility, which gives a bound for the paramagnet-to-ferromagnet transition in a sign-disordered Ising model at sufficiently small fractions of negative bonds.