{"id":358757,"date":"2017-01-27T10:43:50","date_gmt":"2017-01-27T18:43:50","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=358757"},"modified":"2018-10-16T20:03:05","modified_gmt":"2018-10-17T03:03:05","slug":"large-disc-covered-random-walk-n-steps","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/large-disc-covered-random-walk-n-steps\/","title":{"rendered":"How Large A Disc Is Covered By A Random Walk In N Steps?"},"content":{"rendered":"

We show that the largest disc covered by a simple random walk (SRW) on Z<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span> 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<\/span><\/span><\/span><\/span>, the largest disc completely covered at least \u2113<\/span><\/span><\/span><\/span> times by the SRW also has radius n^{1\/4+o(1)}. However, the largest disc completely covered by each of \u2113<\/span><\/span><\/span><\/span> independent simple random walks on Z<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span> after n<\/span><\/span><\/span><\/span> steps is only of radius n<\/span>1<\/span>\/<\/span><\/span><\/span>(<\/span>2<\/span>+<\/span>2<\/span>\u2113<\/span><\/span>\u221a<\/span>)<\/span>+<\/span>o<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>. We complement this by showing that the radius of the largest disc completely covered at least a fixed fraction \u03b1<\/span><\/span><\/span><\/span> of the maximum number of visits to any site during the first n<\/span><\/span><\/span><\/span> steps of the SRW on Z<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>, is n<\/span>(<\/span>1<\/span>\u2212<\/span>\u03b1<\/span><\/span>\u221a<\/span>)<\/span>\/<\/span><\/span><\/span>4<\/span>+<\/span>o<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>. We also show that almost surely, for infinitely many values of n<\/span><\/span><\/span><\/span> it takes about n<\/span>1<\/span>\/<\/span><\/span><\/span>2<\/span>+<\/span>o<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span> 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 Z2 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 […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"The Annals of 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