Winding number correlation for a Brownian loop in a plane

TL;DR

This paper calculates the finite correlation between two winding numbers of a Brownian loop around two marked points in a plane, providing explicit formulas and limits for different separations.

Contribution

It introduces a new explicit calculation of the correlation between two winding numbers in a Brownian loop, extending understanding of their joint behavior.

Findings

Correlation is finite and explicitly computed.

Integral formula depends on scaled separation.

Limits for small and large separations are derived.

Abstract

A Brownian loop is a random walk circuit of infinitely many, suitably infinitesimal, steps. In a plane such a loop may or may not enclose a marked point, the origin, say. If it does so it may wind arbitrarily many times, positive or negative, around that point. Indeed from the (long known) probability distribution, the mean square winding number is infinite, so all statistical moments - averages of powers of the winding number - are infinity (even powers) or zero (odd powers, by symmetry). If an additional marked point is introduced at some distance from the origin, there are now two winding numbers, which are correlated. That correlation, the average of the product of the two winding numbers, is finite and is calculated here. The result takes the form of a single well-convergent integral that depends on a single parameter - the suitably scaled separation of the marked points. The integrals of the correlation weighted by powers of the separation are simple factorial expressions. Explicit limits of the correlation for small and large separation of the marked points are found.