The Characteristic Function of the Renormalized Intersection Local Time of the Planar Brownian Motion

Abstract: In this article we study the distribution of the number of points of a simple random walk, visited a given number of times (the k-multiple point range). In a previous article we had developed a graph theoretical approach which is now extended from the closed to the the non restricted walk. Based on a study of the analytic properties of the first hit determinant a general method to define and calculate the generating functions of the moments of the distribution as analytical functions and express them as an absolutely converging series of graph contributions is given. So a method to calculate the moments for large length in any dimension d > 1 is developed. As an application the centralized moments of the distribution in two dimensions are completely calculated for the closed and the non restricted simple random walk in leading order with all logarithmic corrections of any order. As is well known we therefore have also calculated the characteristic function of the distribution of the renormalized intersection local time of the planar Brownian motion. It turns out to be closely related to the planar Phi-4 theory.