Peierls bounds from Toom contours

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.