Percolation on a product of two trees

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 It relies on an unpublished lemma of Oded Schramm.

Published in at http://dx.doi.org/10.1214/10-AOP618 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• The Alexander-Orbach conjecture holds in high dimensions
• Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk
• Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs
• Percolation on a product of two trees

Cited by in corpus (5)

• Percolation on a product of two trees
• Interacting growth processes and invariant percolation
• Stationary Eden model on groups
• Slightly subcritical hypercube percolation
• Percolation on the product graph of a regular tree and a line does not satisfy the triangle condition at the uniqueness threshold