Open Access
January 2010 The critical Ising model on trees, concave recursions and nonlinear capacity
Robin Pemantle, Yuval Peres
Ann. Probab. 38(1): 184-206 (January 2010). DOI: 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 L3 capacity arises. In particular, on a spherically symmetric tree that has nαbn 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 α≤1/2. Our proofs are based on a new link between nonlinear recursions on trees and Lp capacities.


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Robin Pemantle. Yuval Peres. "The critical Ising model on trees, concave recursions and nonlinear capacity." Ann. Probab. 38 (1) 184 - 206, January 2010.


Published: January 2010
First available in Project Euclid: 25 January 2010

zbMATH: 1197.60092
MathSciNet: MR2599197
Digital Object Identifier: 10.1214/09-AOP482

Primary: 60K35
Secondary: 31C45

Keywords: capacity , Ising model , iteration , nonlinear potential theory , reconstruction , recursion , spin-glass , trees

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 1 • January 2010
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