Localization for Linearly Edge Reinforced Random Walks

TL;DR

This paper investigates the recurrence and transience behavior of linearly edge reinforced random walks (LRRW) on various graphs, establishing phase transitions based on initial weights and graph properties, without relying on the magic formula.

Contribution

It proves recurrence for small weights on bounded degree graphs and transience for large weights on non-amenable graphs, revealing a phase transition in LRRW behavior.

Findings

LRRW is recurrent on bounded degree graphs with small initial weights.

LRRW is transient on non-amenable graphs with large initial weights.

Phase transition in LRRW behavior depending on initial weights and graph type.

Abstract

We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on non-amenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process.