Algebraic and arithmetic area for $m$ planar Brownian paths

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 $<S(m)>\sim \frac{\pi t}{2}\ln m$ 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.

8 pages, 2 figures

References in corpus (3)

• Random Convex Hulls and Extreme Value Statistics
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• Numerical studies of planar closed random walks

Cited by in corpus (3)

• Conformal Correlation Functions in the Brownian Loop Soup
• Statistical Interparticle Potential of an Ideal Gas of Non-Abelian Anyons
• Arithmetic area for m planar Brownian paths