Statistical physics on a product of trees

TL;DR

This paper proves that Bernoulli bond percolation and the Ising model on products of regular trees exhibit second order phase transitions with mean-field exponents, and provides new proofs and generalizations of related decay lemmas.

Contribution

It offers a new proof for phase transitions on products of trees and extends decay results for the critical two-point function.

Findings

Percolation undergoes a second order phase transition with mean-field exponents.

The Ising model on these graphs also exhibits a second order phase transition.

New proof and generalizations of Schramm's decay lemma for the critical two-point function.

Abstract

Let $G$ be the product of finitely many trees $T_1\times T_2 \times \cdots \times T_N$, each of which is regular with degree at least three. We consider Bernoulli bond percolation and the Ising model on this graph, giving a short proof that the model undergoes a second order phase transition with mean-field critical exponents in each case. The result concerning percolation recovers a result of Kozma (2013), while the result concerning the Ising model is new. We also present a new proof, using similar techniques, of a lemma of Schramm concerning the decay of the critical two-point function along a random walk, as well as some generalizations of this lemma.