Oriented Random Walk on the Heisenberg Group and Percolation

TL;DR

This paper demonstrates that oriented random walks on the Heisenberg group have exponential intersection tails, leading to new insights on the transience of percolation clusters on certain transitive graphs with polynomial volume growth.

Contribution

It establishes exponential intersection tails for oriented random walks on the Heisenberg group and applies this to show transience of percolation clusters on specific graphs.

Findings

Oriented random walk on the Heisenberg group has exponential intersection tail.

Percolation clusters are transient on certain transitive graphs with polynomial volume growth.

Results hold for percolation with high retention parameter p close to 1.

Abstract

It is shown that oriented random walk on the Heisenberg group admits exponential intersection tail. As a corollary we get that on any transitive graph of polynomial volume growth, which is not a finite extension of $\mathbb{Z}, \mathbb{Z}^2$, the infinite cluster of percolation with retention parameter $p$, close enough to $1$, is transient.