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

Margherita Disertori, Franz Merkl, Silke W. W. Rolles

Published 2023-05-12Version 1

We show that the vertex-reinforced jump process on the d-dimensional lattice with long-range jumps is transient as long as the initial weights do not decay too fast. The main ingredients in the proof are: the relation of the corresponding random environment on finite boxes with a non-linear supersymmetric hyperbolic sigma model, a comparison with a hierarchical model, and the translation of the hierarchical model to a non-homogeneous effective one-dimensional model.