How large a disc is covered by a random walk in n steps?
Amir Dembo, Yuval Peres, and Jay Rosen
Source: Ann. Probab.
Volume 35, Number 2
(2007), 577-601.
Abstract
We show that the largest disc covered by a simple random walk (SRW) on ℤ2 after n steps has radius n1/4+o(1), thus resolving an open problem of Révész [Random Walk in Random and Non-Random Environments (1990) World Scientific, Teaneck, NJ]. For any fixed ℓ, the largest disc completely covered at least ℓ times by the SRW also has radius n1/4+o(1). However, the largest disc completely covered by each of ℓ independent simple random walks on ℤ2 after n steps is only of radius 
. We complement this by showing that the radius of the largest disc completely covered at least a fixed fraction α of the maximum number of visits to any site during the first n steps of the SRW on ℤ2, is 
. We also show that almost surely, for infinitely many values of n it takes about n1/2+o(1) steps after step n for the SRW to reach the first previously unvisited site (and the exponent 1/2 is sharp). This resolves a problem raised by Révész [Ann. Probab. 21 (1993) 318–328].
Primary Subjects: 60G50
Secondary Subjects: 60G17, 82C41
Keywords: Planar random walk; favorite points; covered discs
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1175287755
Digital Object Identifier: doi:10.1214/009117906000000854
Mathematical Reviews number (MathSciNet):
MR2308589
Zentralblatt MATH identifier:
1123.60026
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