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How large a disc is covered by a random walk in \(n\) steps? (English) Zbl 1123.60026
Consider a simple random walk (SRW) on \({\mathbb Z}^2\) starting at the origin and run for \(n\) steps. The authors ask for the largest disc covered of the SRW without specifying the center of the disc. In the paper the following four main results are given: (1) one concerning the radius of the largest disc completely covered; (2) one if the disc is required to be multiply covered; (3) one if there are \(\ell\) independent SRW; (4) one concerning the random times in which the SRW is sufficiently inside a completely covered disc.
We note that these results solve some problems raised by Révész in 1990 and 1993 [P. Révész, Random walk in random and non-random environments, World Scientific (1990; Zbl 0733.60091), N. J. Teaneck and P. Révész, Ann. Probab. 21, No. 1, 318–328 (1993; Zbl 0770.60034)].

MSC:
60G50 Sums of independent random variables; random walks
60G17 Sample path properties
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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References:
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