  We introduce a new exponential family of probability distributions, which can
be viewed as a multivariate generalization of the Inverse Gaussian
distribution. Considered as the potential of a random Schr\"odinger operator,
this exponential family is related to the random field that gives the mixing
measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing
measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula.
In particular, it yields by direct computation the value of the normalizing
constants of these mixing measures, which solves a question raised by Diaconis.
The results of this paper are instrumental in [Sabot-Zeng,2015], where several
properties of the VRJP and the ERRW are proved, in particular a functional
central limit theorem in transient regimes, and recurrence of the 2-dimensional
ERRW.
