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2021 Power-law decay of weights and recurrence of the two-dimensional VRJP
Gady Kozma, Ron Peled
Author Affiliations +
Electron. J. Probab. 26: 1-19 (2021). DOI: 10.1214/21-EJP639


The vertex-reinforced jump process (VRJP) is a form of self-interacting random walk in which the walker is biased towards returning to previously visited vertices with the bias depending linearly on the local time at these vertices. We prove that, for any initial bias, the weights sampled from the magic formula on a two-dimensional graph decay at least at a power-law rate. Via arguments of Sabot and Zeng, the result implies that the VRJP is recurrent in two dimensions for any initial bias.

Funding Statement

GK is supported by the Israel Science Foundation, by the Jesselson Foundation and by Paul and Tina Gardner. RP is supported by the Israel Science Foundation grants 861/15 and 1971/19 and by the European Research Council starting grant 678520 (LocalOrder).


We are grateful to Thomas Spencer who introduced us to the question of the decay rate of the weights in the VRJP model and suggested that techniques of Mermin–Wagner type may be applicable due to the log-convexity of the determinant. Michael Aizenman explained to RP the idea that fluctuation lower bounds and a priori upper bounds on the field can lead to a proof of decay, as implemented in [25]. We thank Christophe Sabot for encouragement in the writing process.


Download Citation

Gady Kozma. Ron Peled. "Power-law decay of weights and recurrence of the two-dimensional VRJP." Electron. J. Probab. 26 1 - 19, 2021.


Received: 18 May 2020; Accepted: 4 May 2021; Published: 2021
First available in Project Euclid: 16 June 2021

arXiv: 1911.08579
Digital Object Identifier: 10.1214/21-EJP639

Primary: 60K35, 60K37
Secondary: 81T25, 81T60


Vol.26 • 2021
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