English

3-Neighbor bootstrap percolation on two-dimensional grids

Combinatorics 2026-08-06 v1 Probability

Abstract

In the 33-neighbor bootstrap percolation process, a vertex becomes (and remains) infected if at least three of its neighbors are infected. We say that an initial configuration of infected vertices percolates if eventually all vertices are infected. We exactly determine the size of the minimum percolating set for the 33-neighbor bootstrap percolation process on all remaining open cases for rectangular grid graphs PmPnP_m\square P_n. This extends earlier work of Dukes, Noel, and Romer. Additionally, we consider the same question for the toroidal grids CmCnC_m\square C_n, proving upper and lower bounds which are at most one apart and determining the answer precisely in many divisibility cases.

Cite

@article{arxiv.2608.06133,
  title  = {3-Neighbor bootstrap percolation on two-dimensional grids},
  author = {Neal Bushaw and Alexander Clifton},
  journal= {arXiv preprint arXiv:2608.06133},
  year   = {2026}
}

Comments

39 pages, 23 figures.x