Transience of vertex-reinforced jump processes with long-range jumps

Abstract

We show that the vertex-reinforced jump process on the $d$-dimensional lattice with long-range jumps is transient in any dimension $d$ as long as the initial weights do not decay too fast. The main ingredients in the proof are: an analysis of the corresponding random environment on finite boxes, a comparison with a hierarchical model, and the reduction of the hierarchical model to a non-homogeneous effective one-dimensional model. For $d\ge 3$ we also prove transience of the vertex-reinforced jump process with possibly long-range jumps as long as nearest-neighbor weights are large enough.