  This paper concerns the Vertex reinforced jump process (VRJP), the Edge
reinforced random walk (ERRW) and their link with a random Schr\"odinger
operator. On infinite graphs, we define a 1-dependent random potential $\beta$
extending that defined in [20] on finite graphs, and consider its associated
random Schr\"odinger operator $H_\beta$. We construct a random function $\psi$
as a limit of martingales, such that $\psi=0$ when the VRJP is recurrent, and
$\psi$ is a positive generalized eigenfunction of the random Schr\"odinger
operator with eigenvalue $0$, when the VRJP is transient. Then we prove a
representation of the VRJP on infinite graphs as a mixture of Markov jump
processes involving the function $\psi$, the Green function of the random
Schr\"odinger operator and an independent Gamma random variable. On ${{\mathbb
Z}}^d$, we deduce from this representation a zero-one law for recurrence or
transience of the VRJP and the ERRW, and a functional central limit theorem for
the VRJP and the ERRW at weak reinforcement in dimension $d\ge 3$, using
estimates of [10,8]. Finally, we deduce recurrence of the ERRW in dimension $
d=2$ for any initial constant weights (using the estimates of Merkl and Rolles,
[15,17]), thus giving a full answer to the old question of Diaconis. We also
raise some questions on the links between recurrence/transience of the VRJP and
localization/delocalization of the random Schr\"odinger operator $H_\beta$.
