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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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