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

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.