A sharp threshold for a modified bootstrap percolation with recovery

TL;DR

This paper studies a modified bootstrap percolation model on a grid where infected vertices can recover, establishing sharp thresholds for initial infection probabilities that lead to complete infection of the grid.

Contribution

It introduces and analyzes a non-monotone bootstrap percolation model with recovery, providing precise critical probability thresholds for total infection.

Findings

Identifies sharp thresholds for initial infection probability.

Demonstrates conditions for complete infection to occur.

Provides mathematical characterization of the phase transition.

Abstract

Bootstrap percolation is a type of cellular automaton on graphs, introduced as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be in one of two states: `healthy' or `infected' and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this paper, a modification is examined in which infected vertices can return to a healthy state. Vertices are initially infected independently at random and the central question is whether all vertices eventually become infected. The model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy. Sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected.