Abstract
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).
Acknowledgments
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.
Citation
Gady Kozma. Ron Peled. "Power-law decay of weights and recurrence of the two-dimensional VRJP." Electron. J. Probab. 26 1 - 19, 2021. https://doi.org/10.1214/21-EJP639
Information
