A probabilistic cellular automaton that admits no successful basic i.i.d. coupling

TL;DR

This paper demonstrates that for certain probabilistic cellular automata close to deterministic behavior, no successful basic i.i.d. coupling exists, contrasting with monotone PCA where such couplings are always possible.

Contribution

It proves the non-existence of successful basic i.i.d. couplings for a class of ergodic PCA near deterministic limits, highlighting a fundamental difference from monotone PCA.

Findings

No successful basic i.i.d. coupling for p close to 1.

Contrasts with monotone PCA where such couplings exist.

Implications for perfect sampling schemes like CFTP.

Abstract

In this paper, we revisit a classic example of probabilistic cellular automaton (PCA) on {0, 1} Z , namely, addition modulo 2 of the states of the left-and right-neighbouring cells, followed by either preserving the result of the addition, with probability p, or flipping it, with probability 1 -- p. It is well-known that, for any value of p $\in$]0, 1[, this PCA is ergodic. We show that, for p sufficiently close to 1, no coupling of the PCA dynamics based on the composition of i.i.d. random functions of nearest-neighbour states (we call this a basic i.i.d. coupling), can be successful, where successful means that, for any given cell, the probability that every possible initial condition leads to the same state after t time steps, goes to 1 as t goes to infinity. In particular, this precludes the possibility of a CFTP scheme being based on such a coupling. This property stands in sharp contrast with the case of monotone PCA, for which, as soon as ergodicity holds, there exists a successful basic i.i.d. coupling.