We show that the largest disc covered by a simple random walk (SRW) on \\(Z_2\\) after n steps has radius n^{1\/4+o(1)}, thus resolving an open problem of R\\'{e}v\\'{e}sz [Random Walk in Random and Non-Random Environments (1990) World Scientific, Teaneck, NJ]. For any fixed \\(\u2113\\), the largest disc completely covered at least \\(\u2113\\) times by the SRW also has radius n^{1\/4+o(1)}. However, the largest disc completely covered by each of \\(\u2113\\) independent simple random walks on \\(Z_2\\) after \\(n\\) steps is only of radius \\(n_{1\/(2+2\\sqrt{\u2113\u221a})+o(1)}\\). We complement this by showing that the radius of the largest disc completely covered at least a fixed fraction \\(\\alpha\\) of the maximum number of visits to any site during the first \\(n\\) steps of the SRW on \\(Z_2\\), is \\(n_{(1-\\sqrt{\\alpha \u221a})\/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 R\\'{e}v\\'{e}sz [Ann. Probab. 21 (1993) 318–328].<\/p>\n","protected":false},"excerpt":{"rendered":"
We show that the largest disc covered by a simple random walk (SRW) on after n steps has radius n^{1\/4+o(1)}, thus resolving an open problem of R\\'{e}v\\'{e}sz [Random Walk in Random and Non-Random Environments (1990) World Scientific, Teaneck, NJ]. 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