# Polluted Bootstrap Percolation in Three Dimensions

## TL;DR

This paper analyzes the behavior of polluted bootstrap percolation in three dimensions, establishing conditions under which the final occupied density converges to either 0 or 1 as initial probabilities tend to zero.

## Contribution

It provides new bounds on the final density in polluted bootstrap percolation, including a matching power bound for the modified model and improved conditions for the standard model.

## Key findings

- Final density approaches 1 if q is much smaller than p^3(log p^{-1})^{-3}.
- Final density approaches 0 if q exceeds Cp^3 in the modified model.
- Final density approaches 0 if q exceeds Cp^2 in the standard model.

## Abstract

In the polluted bootstrap percolation model, vertices of the cubic lattice $\mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least $3$ occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $p,q\to 0$. We show that this density converges to $1$ if $q \ll p^3(\log p^{-1})^{-3}$ for both standard and modified rules. Our principal result is a complementary bound with a matching power for the modified model: there exists $C$ such that the final density converges to $0$ if $q > Cp^3$. For the standard model, we establish convergence to $0$ under the stronger condition $q>Cp^2$.