Recurrence of the VRJP and Exponential Decay in the \(H^{2|2}\)-Model on the Hierarchical Lattice for \(d\le 2\)

TL;DR

This paper investigates the recurrence and transience of vertex-reinforced jump processes on hierarchical lattices, establishing exponential decay estimates and stability properties for the model, especially in two dimensions.

Contribution

It provides new recurrence results for VRJP on hierarchical lattices and introduces a novel approach using entropy estimates and fractional moments.

Findings

VRJP is recurrent for d<2 and transient for d>2

Exponential decay of fractional moments of Green's function established

Stability under coarse graining crucial for analysis

Abstract

We show that the vertex-reinforced jump processes on a \(d\)-dimensional hierarchical lattice are recurrent for \(d < 2\) and transient for \(d > 2\). We also explore certain regimes when \(d = 2\). The proof of recurrence relies on an exponential decay estimate of the fractional moment of the Green's function, which, unlike the classical approach used for \(\mathbb{Z}^d\), requires additional entropy estimates via stability of the model distribution under coarse grain operation, which leverages its linear reinforcement.