The Annals of Probability
- Ann. Probab.
- Volume 35, Number 2 (2007), 577-601.
How large a disc is covered by a random walk in n steps?
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].
Ann. Probab., Volume 35, Number 2 (2007), 577-601.
First available in Project Euclid: 30 March 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G17: Sample path properties 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]
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