Open Access
January 2008 Counting planar random walk holes
Christian Beneš
Ann. Probab. 36(1): 91-126 (January 2008). DOI: 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 nA(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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Christian Beneš. "Counting planar random walk holes." Ann. Probab. 36 (1) 91 - 126, January 2008.


Published: January 2008
First available in Project Euclid: 28 November 2007

zbMATH: 1151.60020
MathSciNet: MR2370599
Digital Object Identifier: 10.1214/009117907000000204

Primary: 60G50 , 60J65

Keywords: Brownian motion , Random walk

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 1 • January 2008
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