A random polymer approach to the weak disorder phase of the vertex reinforced jump process

In this paper, we study the transient phase of the Vertex Reinforced Jump Process (VRJP) in dimension $d\geq 3$. In Sabot, Zeng (2019), the authors introduce a positive martingale and show that the VRJP is recurrent if and only if that martingale converges to $0$. On $\mathbb{Z}^d$, $d\ge 3$, with constant conductances $W$, it can be shown that there is a critical value $0<W_c(\mathbb{Z}^d)<\infty$, such that the martingale converges to $0$ if $W<W_c(\mathbb{Z}^d)$ or to a positive limit if $W>W_c(\mathbb{Z}^d)$. On the other hand, the VRJP martingale can be interpreted as the partition function of a non-directed polymer with a very specific $1$-dependent random potential. In this paper, we focus on the question of the $L^p$ integrability of the VRJP martingale, which is related to the (diffusive) behavior of the VRJP. First, taking inspiration from the work of Junk (2022) for directed polymers in $\mathbb{Z}^{1+d}$, we prove that on the half-space $\mathbb{H}_d$ of $\mathbb{Z}^d$, for all $W>W_c(\mathbb{H}_d)$ there is some $\delta>0$ such that the VRJP martingale is in $L^{1+\delta}$. Second, we prove that, in dimension $d\geq 4$, the VRJP martingale is in $L^{p}$ for all $p>1$ above the ``slab critical point'' $W_c^{\mathrm{slab}} (\mathbb{Z}^d) = \lim_{m\to\infty} W_c(\mathbb{Z}^{d-1} \times \{-m,\ldots,m\})$. We also propose some related conjectures.