Power-law decay of weights and recurrence of the two-dimensional VRJP

TL;DR

This paper proves that in two-dimensional graphs, the weights of the vertex-reinforced jump process decay at least as a power-law, leading to the recurrence of the process regardless of initial bias.

Contribution

It establishes the power-law decay of weights and recurrence of VRJP in two dimensions, extending understanding of self-interacting random walks.

Findings

Weights decay at least as a power-law in 2D

VRJP is recurrent in 2D for any initial bias

Results rely on the magic formula and Sabot-Zeng arguments

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.