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 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.