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

TL;DR

This paper develops a graph-theoretical approach to analyze the distribution of multiple points visited by planar Brownian motion and simple random walks, providing explicit calculations of moments and the characteristic function.

Contribution

It extends previous methods to non-restricted walks, introduces a new series expansion for moments, and links the characteristic function to planar Phi-4 theory.

Findings

Explicit formulas for moments of the distribution in two dimensions

Calculation of the characteristic function of the renormalized intersection local time

Connection established between the distribution and planar Phi-4 quantum field theory

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.