A 0-1 law for vertex-reinforced random walks on $\mathbb{Z}$ with weight of order $k^\alpha$, $\alpha<1/2$

TL;DR

This paper establishes a clear dichotomy for vertex-reinforced random walks on the integer line with weights growing as a power less than 1/2, showing they are either almost surely recurrent or transient.

Contribution

It proves a 0-1 law for vertex-reinforced random walks with sublinear weights, refining previous results on the set of visited sites.

Findings

The walk is either almost surely recurrent or transient.

The set of sites visited infinitely often is almost surely either empty or infinite.

The result applies to weights of order $k^eta$ with $eta<1/2$.

Abstract

We prove that Vertex Reinforced Random Walk on $\mathbb{Z}$ with weight of order $k^\alpha$, with $\alpha\in [0,1/2)$, is either almost surely recurrent or almost surely transient. This improves a previous result of Volkov who showed that the set of sites which are visited infinitely often was a.s. either empty or infinite.