Monotonicity and phase transition for the VRJP and the ERRW

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.