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

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    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 L3L^3 capacity arises. In particular, on a spherically symmetric tree that has nαbnn^{\alpha}b^n vertices at level nn (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 α≤1/2\alpha\le1/2. Our proofs are based on a new link between nonlinear recursions on trees and LpL^p capacities.Comment: Published in at http://dx.doi.org/10.1214/09-AOP482 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
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