# Polluted Bootstrap Percolation with Threshold Two in All Dimensions

## TL;DR

This paper studies a bootstrap percolation model on high-dimensional cubic lattices, showing that the occupied site density approaches 1 as initial occupation and closure probabilities go to zero, resolving part of a conjecture.

## Contribution

It proves that in all dimensions d≥3, the final occupied density tends to 1 as p,q→0, regardless of their scaling, partially resolving Morris's conjecture.

## Key findings

- Final density approaches 1 as p,q→0
- Contrasts with the 2D case where the critical parameter differs
- Provides new insights into high-dimensional bootstrap percolation

## Abstract

In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice Z^d of dimension d>=3 with threshold r=2, we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resolves a conjecture of Morris, and contrasts with the d=2 case, where Gravner and McDonald proved that the critical parameter is q/{p^2}.