Growing in time IDLA cluster is recurrent

TL;DR

This paper proves that in dimensions three and higher, the IDLA cluster's structure ensures the recurrence of simple random walks on it, by establishing near optimal Cheeger constants and heat kernel bounds.

Contribution

It demonstrates that large IDLA clusters in high dimensions have near optimal Cheeger constants, leading to recurrence results for random walks on these clusters.

Findings

IDLA clusters have near optimal Cheeger constants when large

Random walks on growing IDLA clusters are recurrent in dimensions ≥ 3

Heat kernel bounds imply recurrence for simple random walks

Abstract

We show that Internal Diffusion Limited Aggregation (IDLA) on $\mathbb{Z}^d$ has near optimal Cheeger constant when the growing cluster is large enough. This implies, through a heat kernel lower bound derived previously in [H], that simple random walk evolving independently on growing in time IDLA cluster is recurrent when $d\ge 3$.