Almost sure behavior of linearly edge-reinforced random walks on the half-line

TL;DR

This paper investigates the long-term behavior of linearly edge-reinforced random walks on the positive half-line, revealing phase transitions in speed depending on parameters and extending classical limit theorems.

Contribution

It provides new almost sure limit theorems and bounds for the walk's trajectory, highlighting phase transitions in the recurrent regime based on reinforcement parameters.

Findings

Walk speed is slower with reinforcement than without.

Phase transition at =2 in the critical case =1.

Limit theorem analogous to the law of the iterated logarithm.

Abstract

We study linearly edge-reinforced random walks on $\mathbb{Z}_+$, where each edge $\{x,x+1\}$ has the initial weight $x^{\alpha} \vee 1$, and each time an edge is traversed, its weight is increased by $\Delta$. It is known that the walk is recurrent if and only if $\alpha \leq 1$. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For $\alpha<1$ and $\Delta>0$, we obtain a limit theorem which is a counterpart of the law of the iterated logarithm for simple random walks. This reveals that the speed of the walk with $\Delta>0$ is much slower than $\Delta=0$. In the critical case $\alpha=1$, our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at $\Delta=2$.