## The Annals of Probability

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

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

Robin Pemantle and Yuval Peres

#### 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 *L*^{3} capacity arises. In particular, on a spherically symmetric tree that has *n*^{α}*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 *α*≤1/2. Our proofs are based on a new link between nonlinear recursions on trees and *L*^{p} 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