Bootstrap percolation, probabilistic cellular automata and sharpness

TL;DR

This paper establishes new connections between percolation, bootstrap percolation, and probabilistic cellular automata, proving sharp phase transitions and linking classical stability results to bootstrap percolation universality.

Contribution

It introduces novel links between these models, proves phase transition sharpness for certain automata and bootstrap models, and relates classical stability results to bootstrap percolation.

Findings

Proved sharp phase transition of attractive absorbing probabilistic cellular automata.

Established sharpness of bootstrap percolation models and kinetically constrained models.

Connected Toom's stability result to bootstrap percolation universality.

Abstract

We establish new connections between percolation, bootstrap percolation, probabilistic cellular automata and deterministic ones. Surprisingly, by juggling with these in various directions, we effortlessly obtain a number of new results in these fields. In particular, we prove the sharpness of the phase transition of attractive absorbing probabilistic cellular automata, a class of bootstrap percolation models and kinetically constrained models. We further show how to recover a classical result of Toom on the stability of cellular automata w.r.t. noise and, inversely, how to deduce new results in bootstrap percolation universality from his work.