Algebraic and arithmetic area for $m$ planar Brownian paths

TL;DR

This paper analyzes the asymptotic behavior of the average arithmetic area enclosed by multiple independent planar Brownian paths, revealing a logarithmic growth with the number of paths and providing a closed-form distribution for the algebraic area.

Contribution

It derives the leading and subleading terms of the average arithmetic area for many Brownian paths and presents a closed-form expression for the algebraic area distribution.

Findings

Average arithmetic area scales as (π t/2) ln m for large m

Subleading contributions from the 0-winding sector are identified

Closed-form expression for algebraic area distribution is provided

Abstract

The leading and next to leading terms of the average arithmetic area $< S(m)>$ enclosed by $m\to\infty$ independent closed Brownian planar paths, with a given length $t$ and starting from and ending at the same point, is calculated. The leading term is found to be $< S(m) > \sim {\pi t\over 2}\ln m$ and the $0$-winding sector arithmetic area inside the $m$ paths is subleading in the asymptotic regime. A closed form expression for the algebraic area distribution is also obtained and discussed.