Polynomial localization of the 2D-Vertex Reinforced Jump Process

Abstract

We prove polynomial decay of the mixing field of the Vertex Reinforced Jump Process (VRJP) on ${\mathbb {Z}}^{2}$ with bounded conductances. Using [22] we deduce that the VRJP on ${\mathbb {Z}}^{2}$ with any constant conductances is almost surely recurrent. It gives a counterpart of the result of Merkl, Rolles [16] and Sabot, Zeng [22] for the 2-dimensional Edge Reinforced Random Walk.