Maximum percolation time in two-dimensional bootstrap percolation

TL;DR

This paper determines the maximum time for a bootstrap percolation process on an n-by-n grid to infect all vertices, establishing it as approximately 13/18 of n squared, advancing extremal results in the field.

Contribution

It proves a new upper bound on the maximum percolation time in two-dimensional bootstrap percolation, refining previous extremal results.

Findings

Maximum percolation time is approximately 13n^2/18+O(n).

Established a new upper bound for infection spread time.

Contributes to extremal combinatorics in bootstrap percolation.

Abstract

We consider a classic model known as bootstrap percolation on the $n \times n$ square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least $2$ already infected neighbours. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap percolation process can take to eventually infect the entire vertex set of the grid is $13n^2/18+O(n)$.