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 $x^{\alpha }\vee 1$, 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 $\alpha \le 1$. 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 $\Delta =2$.

18 pages, with minor updates

References in corpus (1)

• Logarithmic speeds for one-dimensional perturbed random walk in random environment