Arithmetic area for m planar Brownian paths

TL;DR

This paper analyzes the arithmetic area enclosed by multiple independent planar Brownian paths, focusing on the distribution of points with specified winding numbers, especially in the limit of many paths, utilizing SLE insights.

Contribution

It extends previous work by examining the arithmetic area for m independent Brownian paths and explores asymptotic behaviors using SLE techniques.

Findings

Derived asymptotic results for large m

Connected winding sector areas with SLE properties

Provided insights into the distribution of winding sectors

Abstract

We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm times by path m. Various results are obtained in the asymptotic limit m->infinity. A key observation is that, since the paths are independent, one can use in the m paths case the SLE information, valid in the 1-path case, on the 0-winding sectors arithmetic area.