IDLA on the Supercritical Percolation Cluster

TL;DR

This paper demonstrates that the IDLA process on the supercritical percolation cluster behaves similarly to the Euclidean lattice, with the aggregate covering a Euclidean ball almost surely, confirming the robustness of IDLA behavior in random environments.

Contribution

It proves that IDLA on the supercritical percolation cluster exhibits Euclidean-like growth, extending understanding of IDLA in random media.

Findings

IDLA covers Euclidean balls almost surely on the cluster

The ratio of covered vertices to particles approaches one

Behavior matches that on regular Euclidean lattices

Abstract

We consider the internal diffusion limited aggregation (IDLA) process on the infinite cluster in supercritical Bernoulli bond percolation on Euclidean lattices. It is shown that the process on the cluster behaves like it does on the Euclidean lattice, in that the aggregate covers all the vertices in a Euclidean ball around the origin, such that the ratio of vertices in this ball to the total number of particles sent out approaches one almost surely.