Explosive appearance of cores and bootstrap percolation on lattices

TL;DR

This paper proves that in bootstrap percolation on 2D lattices, the transition from almost no infected vertices to complete infection occurs almost instantaneously, using new methods to handle complex infection rules.

Contribution

The authors develop a novel technique to analyze bootstrap percolation on 2D lattices, extending results to a broad class of infection rules beyond standard methods.

Findings

The size of the 3-core jumps from 0 to nearly n in a single step.

Infection spread on 2D lattices exhibits an abrupt phase transition.

New analytical methods enable results for complex neighborhood rules.

Abstract

Consider the process where the $n$ vertices of a square $2$-dimensional torus appear consecutively in a random order. We show that typically the size of the $3$-core of the corresponding induced unit-distance graph transitions from $0$ to $n-o(n)$ within a single step. Equivalently, by infecting the vertices of the torus in a random order, under two-neighbour bootstrap percolation, the size of the infected set transitions instantaneously from $o(n)$ to $n$. This hitting time result answers a question of Benjamini. We also study the much more challenging and general setting of bootstrap percolation on two-dimensional lattices for a variety of finite-range infection rules. In this case, powerful but fragile bootstrap percolation tools such as the rectangles process and the Aizenman-Lebowitz lemma become unavailable. We develop a new method complementing and replacing these standard techniques, thus allowing us to prove the above hitting time result for a wide family of threshold bootstrap percolation rules on the $2$-dimensional square lattice, including neighbourhoods given by large $\ell^p$ balls for $p\in[1,\infty]$.