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

Abstract

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