Summary: 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 Schrödinger 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 [the first and third authors, “A random Schrödinger operator associated with the vertex reinforced jump process on infinite graphs”, Preprint, arXiv:1507.07944], 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.