Counting planar random walk holes

Details Counting planar random walk holes Counting planar random walk holes

2006-11-06

arxiv.org/abs/math/0611144v2

Annals of Probability 2008, Vol. 36, No. 1, 91-126

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117907000000204 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117907000000204

We study two variants of the notion of holes formed by planar simple random walk of time duration $2n$ and the areas associated with them. We prove in both cases that the number of holes of area greater than $A(n)$, where $\{A(n)\}$ is an increasing sequence, is, up to a logarithmic correction term, asymptotic to $n\cdot A(n)^{-1}$ for a range of large holes, thus confirming an observation by Mandelbrot. A consequence is that the largest hole has an area which is logarithmically asymptotic to $n$. We also discuss the different exponent of 5/3 observed by Mandelbrot for small holes.

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