Percolation on a product of two trees

TL;DR

This paper proves that critical percolation on a product of two regular trees with degree at least 3 satisfies the triangle condition, using a novel non-perturbative approach based on an unpublished lemma.

Contribution

It establishes the triangle condition for critical percolation on a product of two trees without degree restrictions, introducing a new proof technique.

Findings

Critical percolation on the product of two regular trees satisfies the triangle condition.

The proof is non-perturbative and does not depend on vertex degrees.

Utilizes an unpublished lemma of Oded Schramm.

Abstract

We show that critical percolation on a product of two regular trees of degree $\ge$ 3 satisfies the triangle condition. The proof does not examine the degrees of vertices and is not "perturbative" in any sense. It relies on an unpublished lemma of Oded Schramm.