Substitution Markov chains and Martin boundaries

TL;DR

This paper investigates the Martin boundaries of substitution Markov chains, revealing differences in metric structure despite topological similarities, and offers insights into their probabilistic and topological properties.

Contribution

It provides a novel analysis of Martin boundaries for substitution Markov chains, highlighting differences between topological and metric descriptions.

Findings

Martin boundaries can be topologically similar but metrically different

An example shows boundary homeomorphic to coin tossing process with distinct metric properties

Enhances understanding of probabilistic and topological aspects of substitution Markov chains

Abstract

Substitution Markov chains have been introduced [7] as a new model to describe molecular evolution. In this note, we study the associated Martin boundaries from a probabilistic and topological viewpoint. An example is given that, although having a boundary homeomorphic to the well-known coin tossing process, has a metric description that differs significantly.