Explicit estimates in the Bramson-Kalikow model

TL;DR

This paper provides explicit parameter estimates for phase transitions in the Bramson-Kalikow model by developing a new criterion for non-uniqueness of g-measures, using $ar{d}$-distance estimates between Markov chains.

Contribution

It introduces a simple, new criterion for non-uniqueness of g-measures and applies it to explicitly identify phase transition parameters in the Bramson-Kalikow model.

Findings

Explicit parameter estimates for phase transition in the Bramson-Kalikow model.

A new criterion for non-uniqueness of g-measures based on $ar{d}$-distances.

Optimal method for binary regular attractive functions.

Abstract

The aim of the present article is to explicitly compute parameters for which the Bramson-Kalikow model exhibits phase-transition. The main ingredient of the proof is a simple new criterion for non-uniqueness of $g$-measures. We show that the existence of multiple $g$-measures compatible with a function $g$ can be proved by estimating the $\bar{d}$-distances between some suitably chosen Markov chains. The method is optimal for the important class of binary regular attractive functions, which includes the Bramson-Kalikow model.