The critical Ising model on trees, concave recursions and nonlinear capacity

Details The critical Ising model on trees, concave recursions and nonlinear capacity The critical Ising model on trees, concave recursions and nonlinear capacity

2005-03-08

arxiv.org/abs/math/0503137v3

Annals of Probability 2010, Vol. 38, No. 1, 184-206

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP482 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP482

We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an $L^3$ capacity arises. In particular, on a spherically symmetric tree that has $n^{\alpha}b^n$ vertices at level $n$ (up to bounded factors), we prove that there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature if and only if $\alpha\le1/2$. Our proofs are based on a new link between nonlinear recursions on trees and $L^p$ capacities.

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