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

2018/04/08 by Collevecchio, Andrea, Zeng, Xiaolin · 1 citation
#60K35 #60K37 (Primary) #81T25 #81T60 (Secondary) #82B44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1804.02739

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.

Cited by

• VRJP recurrence and fractional-moment decay for the \(H^2|2\) model's effective field on the hierarchical lattice

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