# The edge reinforcement process and the Disertori-Spencer-Zinbauer integral

Let $M_x$ be a symmetric matrix of order $n$ with fixed non positive off- diagonal coefficients $-w_{ij}$ and with diagonal $(2x_1,ldots,2x_n).$ We calculate for $ b_1>0,ldots b_n>0$ the integral $GST_n(y)=int exp left(-langle x,yrangle -frac{1}{2}b^*M_x^{-1}bright)(det M_x)^{-1/2}dx.$ The domain of integration is the part of $mathbb{R}^n$ for which $M_x$ is positive definite. The result is simple although the proof is involved. This is a less daunting reformulation of a Disertori-Spencer-Zinbauer integral. These integrals occur in the study made by Sabot and Tarres of the reinforced Markov chain on a graph when the edges are the $(ij)'$s such that $w_{ij}>0.$. In this non homogeneous chain, the more you use an edge, the more the probability to use it again in the future increases. This creates a family of distributions on $mathbb{R}^n$ with striking properties like stability by marginalization and (up to a translation) stability by conditioning. This is joint work with Jacek Wesolowski.