A note on recurrence of the Vertex reinforced jump process and fractional moments localization

Abstract

We give a simple proof for recurrence of vertex reinforced jump process on ${\mathbb{Z}}^d$, under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on ${\mathbb{Z}}^d$ is recurrent for strong reinforcement. Finally, we prove that the $H^{(2|2)}$ model on ${\mathbb{Z}}^d$ localizes at strong disorder. Even though these results are well-known, we propose a unified approach, which also has the advantage to provide shorter proofs, and relies on estimating fractional moments, introduced by Aizenman and Molchanov.