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 n^1/4+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 n^1/4+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 $n^{1/(2+2\sqrt{\ell})+o(1)}$. 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 n^1/2+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].