Lamplighter Random Walks and Entropy-Sensitivity of Languages

TL;DR

This thesis explores the relationship between geometric properties of infinite graphs and probabilistic objects like random walks, focusing on lamplighter graphs and entropy sensitivity of languages derived from Markov chains.

Contribution

It provides new insights into the convergence, harmonic functions, and Poisson boundaries of lamplighter random walks, and introduces entropy sensitivity analysis of languages from directed graphs.

Findings

Convergence and Poisson boundary characterized for lamplighter random walks.

Analysis of entropy sensitivity in languages associated with Markov chains.

Geometric methods applied to study random walks on wreath product graphs.

Abstract

The main purpose of this thesis is to study the interplay between geometric properties of infinite graphs and analytic and probabilistic objects such as transition operators, harmonic functions and random walks on these graphs. For a transient random walk, there are several problems one is interested in: for instance to study its convergence, to describe the bounded harmonic functions for the random walk, to describe its Poisson boundary, or to study the parameter of exponential decay of the transition probabilities of the random walk. In the first part of the thesis we deal with similar problems in the context of random walks on the so-called lamplighter graphs, which are wreath products of graphs. The convergence and the Poisson boundary of lamplighter random walks is studied for different underlying graphs, and the used methods are mostly of a geometrical nature. In the second part of the thesis we consider Markov chains on directed, labelled graphs. With such graphs we associate in a natural way a class of infinite languages (sets of labels of paths in the graph) and we study the growth sensitivity (or entropy sensitivity) of these languages using Markov chains.