We give a simple proof for recurrence of vertex reinforced jump process on , under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on is recurrent for strong reinforcement. Finally, we prove that the model on 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.
A.C. was supported by ARC grant DP180100613 and Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS) CE140100049. X.Z. was supported by ERC starting grant 678520 and supported in part by Israel Science Foundation grant 861/15.
X. Z. want to thank R. Peled for his encouragement of writing this paper. We thank an anonymous referee for very helpful comments.
"A note on recurrence of the Vertex reinforced jump process and fractional moments localization." Electron. J. Probab. 26 1 - 16, 2021. https://doi.org/10.1214/21-EJP609