Monotonicity and phase transition for the VRJP and the ERRW

TL;DR

This paper demonstrates that increasing initial weights in VRJP and ERRW models on  lattices makes them more transient, establishing a monotonicity property and linking recurrence to electrical network models, thus confirming a unique phase transition.

Contribution

It proves monotonicity of recurrence and transience in VRJP and ERRW models and connects recurrence to electrical network behavior.

Findings

Increasing initial weights increases transience.

Recurrence is linked to electrical network recurrence.

Phase transition is unique and well-defined.

Abstract

The vertex-reinforced jump process (VRJP), introduced by Davis and Volkov, is a continuous-time process that tends to come-back to already visited vertices. It is closely linked to the edge-reinforced random walk (ERRW) introduced by Coppersmith and Diaconis in 1986 which is more likely to cross edges it has already crossed. On $\mathbb{Z}^d$ for $d\geq 3$, both models where shown to be recurrent for small enough initial weights (by Sabot, Tarr\`es(2015) and Angel,Crawford,Kozma(2014)) and transient for large enough initial weights (by Disertori,Sabot,Tarr\`es(2015) and Sabot,Tarr\`es(2015)). We show through a coupling of the VRJP for different weights that the VRJP (and the ERRW) exhibits some monotonicity. In particular, we show that increasing the initial weights of the VRJP and the ERRW makes them more transient which means that the recurrence/transience phase transition is necessarily unique. Furthermore, by making the weights go to infinity, we show that the recurrence of the ERRW and the VRJP is implied by the recurrence of a random walk in deterministic electrical network.