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 Z d , under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on Z d is recurrent for strong reinforcement. Finally, we prove that the H (2|2) model on 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.