Variations on Reinforced Random Walks

TL;DR

This thesis explores modifications to edge-reinforced random walks, analyzing their recurrence, behavior with multiple walkers, and effects of bias, revealing phase transitions and convergence properties in various settings.

Contribution

It introduces new results on multi-walker interactions, convergence of edge weights, and phase transitions caused by bias in reinforced random walks.

Findings

Edge weights on three nodes resemble a Pólya urn.

On Z, multiple walkers are either all recurrent or all finite-range.

Bias can induce a phase transition between recurrence and transience.

Abstract

This thesis examines edge-reinforced random walks with some modifications to the standard definition. An overview of known results relating to the standard model is given and the proof of recurrence for the standard linearly edge-reinforced random walk on bounded degree graphs with small initial edge weights is repeated. Then, the edge-reinforced random walk with multiple walkers influencing each other is considered. The following new results are shown: on a segment of three nodes, the edge weights resemble a P\'olya urn and the fraction of the edge weights divided by the total weight forms a converging martingale. On Z, the behavior is the same as for a single walker - either all walkers have finite range or all walkers are recurrent. Finally, edge-reinforced random walks with a bias in a certain direction are analysed, in particular on Z. It is shown that the bias can introduce a phase transition between recurrence and transience, depending on the strength of the bias, thus fundamentally altering the behavior in comparison to the standard linearly reinforced random walk.