Vertex-reinforced random walk on Z with sub-square-root weights is recurrent

TL;DR

This paper proves that vertex-reinforced random walks on the integers with weights growing slower than the square root are recurrent, confirming a conjecture for certain weights, while the case for higher weights remains unresolved.

Contribution

It establishes recurrence for vertex-reinforced random walks with weights of order k^alpha for alpha in [0, 1/2), confirming a conjecture and highlighting open problems for larger alpha.

Findings

Recurrent behavior for alpha in [0, 1/2)

Confirmation of Volkov's conjecture in this range

Open problem for alpha in [1/2, 1)

Abstract

We prove that vertex-reinforced random walk on the integers with weight of order k to the power alpha, for alpha in [0, 1/2), is recurrent. This confirms a conjecture of Volkov for alpha<1/2. The conjecture for alpha in [1/2, 1) remains open.