Seven-dimensional forest fires

TL;DR

This paper investigates high-dimensional Bernoulli percolation, demonstrating that removing a thinner infinite cluster from a thick one still results in an infinite component, with implications for high-dimensional forest fire models.

Contribution

It reveals that in high dimensions, the removal of a thinner infinite cluster from a thicker one preserves an infinite cluster, advancing understanding of self-destructive percolation.

Findings

Removing a thinner infinite cluster leaves an infinite component in high-dimensional Bernoulli percolation.

Implications for the van den Berg-Brouwer forest fire process in high dimensions.

Provides insights into the structure of clusters in high-dimensional percolation models.

Abstract

We show that in high dimensional Bernoulli percolation, removing from a thin infinite cluster a much thinner infinite cluster leaves an infinite component. This observation has implications for the van den Berg-Brouwer forest fire process, also known as self-destructive percolation, for dimension high enough.