On polluted bootstrap percolation in Cartesian grids

TL;DR

This paper investigates a variant of bootstrap percolation on grid graphs where some vertices are permanently non-infectious, providing formulas and bounds for the minimum initial infection needed for complete percolation.

Contribution

It introduces and analyzes an extremal polluted bootstrap percolation model on Cartesian grids, deriving a closed formula for the minimum 2-neighbor percolation number.

Findings

Established a closed formula for the minimum 2-neighbor percolation number on polluted grids.

Derived a lower bound for the percolation number in the case of maximum pollution.

Extended understanding of bootstrap percolation in polluted environments.

Abstract

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we continue the study of polluted bootstrap percolation introduced and studied by Gravner and McDonald [Bootstrap percolation in a polluted environment. J.\ Stat\ Physics 87 (1997) 915--927] where in this variant some vertices are permanently in the non-infected state. We study an extremal (combinatorial) version of the bootstrap percolation problem in a polluted environment, where our main focus is on the class of grid graphs, that is, the Cartesian product $P_m \square P_n$ of two paths $P_m$ and $P_n$ on $m$ and $n$ vertices, respectively. Given a number of polluted vertices in a Cartesian grid we establish a closed formula for the minimum $2$-neighbor bootstrap percolation number of the polluted grid, and obtain a lower bound for the other extreme.