Uniqueness of the infinite cluster for monotone percolation models without insertion tolerance

TL;DR

This paper proves the almost sure uniqueness of the infinite cluster in certain dependent percolation models on bZ^d, including the Abelian sandpile and bootstrap percolation, under specific conditions.

Contribution

It establishes the uniqueness of the infinite cluster in a broad class of dependent models without relying on standard conditions like finite energy.

Findings

Supercritical phase contains a unique infinite cluster under specified conditions.

The result applies to models like Abelian sandpile, activated random walk, and bootstrap percolation.

Answers a question about percolation of toppled vertices in the Abelian sandpile.

Abstract

We consider a broad class of dependent site-percolation models on $\mathbb{Z}^d$ obtained by applying a monotone automaton to a random initial particle configuration drawn from a stochastically increasing family of measures. We prove that whenever the underlying particle configuration is sampled from an insertion-tolerant measure and the avalanches generated by the dynamics produce connected sets, the supercritical phase almost surely contains a unique infinite cluster. Our result applies to several well-studied interacting particle systems, including the Abelian sandpile, activated random walk, and bootstrap percolation. In these models, the induced percolation measure typically does not satisfy standard conditions such as finite energy or insertion tolerance, so the classical Burton-Keane argument does not apply. As an application, we answer a question of Fey, Meester, and Redig concerning percolation of the toppled vertices in the Abelian sandpile with i.i.d. initial configuration.