THE CRITICAL ISING MODEL ON TREES, CONCAVE RECURSIONS AND NONLINEAR CAPACITY

• 成果类型: Article
• 署名作者: Pemantle, Robin; Peres, Yuval
• 署名单位: University of Pennsylvania; Microsoft
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP482
• 发表日期: 2010
• 页码: 184-206
• 关键词: disordered state; glauber dynamics; bethe lattice; general trees; random-walks; percolation; extremality; graphs

摘要

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 <= 1/2. Our proofs are based on a new link between nonlinear recursions on trees and L-p capacities.