# Polynomial localization of the 2D-Vertex Reinforced Jump Process

## TL;DR

This paper proves polynomial decay of the mixing field for the 2D Vertex Reinforced Jump Process with bounded conductances, establishing recurrence and extending results from edge-reinforced random walks.

## Contribution

It demonstrates polynomial decay of the mixing field for VRJP on  with bounded conductances, providing new insights into its recurrence properties.

## Key findings

- VRJP on  with bounded conductances exhibits polynomial decay.
- VRJP on  with any constant conductances is almost surely recurrent.
- Results extend understanding of reinforced random walks in two dimensions.

## Abstract

We prove polynomial decay of the mixing field of the Vertex Reinforced Jump Process (VRJP) on $\Bbb{Z}^2$ with bounded conductances. Using [17] we deduce that the VRJP on $\Bbb{Z}^2$ with any constant conductances is almost surely recurrent. It gives a counterpart of the result of Merkl, Rolles [14] and Sabot, Zeng [17] for the 2-dimensional Edge Reinforced Random Walk.