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March 2007 How large a disc is covered by a random walk in n steps?
Amir Dembo, Yuval Peres, Jay Rosen
Ann. Probab. 35(2): 577-601 (March 2007). DOI: 10.1214/009117906000000854


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 , the largest disc completely covered at least times by the SRW also has radius n1/4+o(1). However, the largest disc completely covered by each of independent simple random walks on ℤ2 after n steps is only of radius $n^{1/(2+2\sqrt{\ell})+o(1)}$. 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 $n^{(1-\sqrt{\alpha})/4+o(1)}$. 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].


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Amir Dembo. Yuval Peres. Jay Rosen. "How large a disc is covered by a random walk in n steps?." Ann. Probab. 35 (2) 577 - 601, March 2007.


Published: March 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1123.60026
MathSciNet: MR2308589
Digital Object Identifier: 10.1214/009117906000000854

Primary: 60G50
Secondary: 60G17 , 82C41

Keywords: covered discs , favorite points , Planar random walk

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.35 • No. 2 • March 2007
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