In this talk, we consider the Multivariate inverse Gaussian distribution on the Dyson hierarchical lattice as a random Schrödinger operator. We will demonstrate how a coarse-graining/fine-graining property of the model can be used to perform renormalization group analysis in the position space, in particular, at the critical dimension 2, we construct the fine-mesh limit (in the Ultra-violet) and we discuss some relations to the Vertex reinforced jump process on the Dyson lattice.