Algebraic and arithmetic area for $m$ planar Brownian paths

Details Algebraic and arithmetic area for $m$ planar Brownian paths Algebraic and arithmetic area for $m$ planar Brownian paths

2011-01-21

arxiv.org/abs/1101.4135v1

Physics

Mathematical Physics

8 pages, 2 figures

Scientific paper

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.

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