The Annals of Probability

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

Robin Pemantle and Yuval Peres

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Probab., Volume 38, Number 1 (2010), 184-206.

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1264433996

Digital Object Identifier
doi:10.1214/09-AOP482

Mathematical Reviews number (MathSciNet)
MR2599197

Zentralblatt MATH identifier
1197.60092

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 31C45: Other generalizations (nonlinear potential theory, etc.)

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

Citation

Pemantle, Robin; Peres, Yuval. The critical Ising model on trees, concave recursions and nonlinear capacity. Ann. Probab. 38 (2010), no. 1, 184--206. doi:10.1214/09-AOP482. https://projecteuclid.org/euclid.aop/1264433996


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