Linearly Edge-Reinforced Random Walks

TL;DR

This thesis studies linearly edge-reinforced random walks on infinite and random trees, characterizing their recurrence and transience, identifying phase transitions, and exploring their behavior on Z with stationary distributions.

Contribution

It provides a comprehensive analysis of recurrence, transience, and phase transitions for edge-reinforced random walks on various trees, including new results on their limiting distributions on Z.

Findings

Recurrence and transience are characterized by the branching number and reinforcement parameter.

A phase transition from transience to recurrence occurs at a critical parameter value.

Existence of stationary distributions with finite moments for walks on Z.

Abstract

This thesis examines linearly edge-reinforced random walks on infinite trees. In particular, recurrence and transience of such random walks on general (fixed) trees as well as on Galton-Watson trees (i.e. random trees) is characterized, and shown to be related to the branching number of these trees and a so-called reinforcement parameter. A phase transition from transience to recurrence takes place at a critical parameter value. As a tool, random walks in random environment are introduced and known results are repeated, together with detailed proofs. A result on quasi-independent percolation is proved as a by-product. Finally, for the edge-reinforced random walk on Z, the existence of a kind of stationary / limiting distribution with finite moments is shown.