## Annals of Probability

### How large a disc is covered by a random walk in n steps?

#### 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 , 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].

#### Article information

Source
Ann. Probab., Volume 35, Number 2 (2007), 577-601.

Dates
First available in Project Euclid: 30 March 2007

https://projecteuclid.org/euclid.aop/1175287755

Digital Object Identifier
doi:10.1214/009117906000000854

Mathematical Reviews number (MathSciNet)
MR2308589

Zentralblatt MATH identifier
1123.60026

#### Citation

Dembo, Amir; Peres, Yuval; Rosen, Jay. How large a disc is covered by a random walk in n steps?. Ann. Probab. 35 (2007), no. 2, 577--601. doi:10.1214/009117906000000854. https://projecteuclid.org/euclid.aop/1175287755

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