Sharp metastability threshold for an anisotropic bootstrap percolation model

TL;DR

This paper proves the existence of a sharp metastability threshold in an anisotropic bootstrap percolation model, marking the first rigorous demonstration of such a threshold in this class of models.

Contribution

It provides the first mathematical proof of a sharp metastability threshold for an anisotropic bootstrap percolation model.

Findings

Establishes a sharp metastability threshold for the model

First rigorous proof for anisotropic bootstrap percolation threshold

Identifies critical probability behavior in the model

Abstract

Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" bootstrap percolation model: the neighborhood of a point (m,n) is the set \[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\] At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.