Peierls bounds from Toom contours

TL;DR

This paper simplifies Toom's Peierls argument to analyze the stability of monotone cellular automata, including those with intrinsic randomness, and establishes bounds on critical parameters.

Contribution

It provides a simplified Peierls argument framework and applies it to prove stability results for cellular automata with intrinsic randomness.

Findings

Proves stability of certain cellular automata with intrinsic randomness.

Derives lower bounds on critical parameters for deterministic cellular automata.

Simplifies the proof technique for Toom's Peierls argument.

Abstract

For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. The proof of sufficiency is based on an intricate Peierls argument. We present a simplified version of this Peierls argument. Our main motivation is the open problem of determining stability of monotone cellular automata with intrinsic randomness, in which for the unperturbed evolution the local update rules at different space-time points are chosen in an i.i.d. fashion according to some fixed law. We apply Toom's Peierls argument to prove stability of a class of cellular automata with intrinsic randomness and also derive lower bounds on the critical parameter for some deterministic cellular automata.