Attractive regular stochastic chains: perfect simulation and phase transition

TL;DR

This paper investigates the conditions under which attractive regular stochastic chains have unique stationary distributions, linking this to finitary codings and exponential concentration, and explores phase transition phenomena.

Contribution

It establishes equivalences between uniqueness, finitary coding, and measure concentration for attractive regular chains, and provides new results on phase transitions for infinite order chains.

Findings

Uniqueness of the stationary chain is equivalent to finitary coding or exponential measure concentration.

Chains with continuous, attractive kernels can be sampled via coupling-from-the-past.

For the Bramson-Kalikow model, uniqueness implies finitary coding of a finite alphabet i.i.d. process.

Abstract

We prove that uniqueness of the stationary chain, or equivalently, of the $g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an i.i.d. process with countable alphabet, (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson-Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.