zbMATH — the first resource for mathematics

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)].

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
PDF BibTeX Cite
Full Text: DOI arXiv
[1] Daviaud, O. (2006). Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34 962–982. · Zbl 1104.60062
[2] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2001). Thick points for planar Brownian motion and the Erdős–Taylor conjecture on random walk. Acta Math. 186 239–270. · Zbl 1008.60063
[3] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2004). Cover time for Brownian motion and random walks in two dimensions. Ann. of Math. (2) 160 433–464. · Zbl 1068.60018
[4] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2006). Late points for random walks in two dimensions. Ann. Probab. 34 219–263. · Zbl 1100.60057
[5] Erdős, P. and Révész, P. (1991). Three problems on the random walk in \(\mathbbZ^d\). Studia Sci. Math. Hungar. 26 309–320. · Zbl 0774.60036
[6] Kahane, J.-P. (1985). Some Random Series of Functions , 2nd ed. Cambridge Univ. Press. · Zbl 0571.60002
[7] Lawler, G. (1991). Intersections of Random Walks . Birkhäuser, Boston, MA. · Zbl 1228.60004
[8] Révész, P. (1990). Random Walk in Random and Non-Random Environments . World Scientific, Teaneck, NJ. · Zbl 0733.60091
[9] Révész, P. (1993). Clusters of a random walk in the plane. Ann. Probab. 21 318–328. · Zbl 0770.60034
[10] Rosen, J. (2005). A random walk proof of the Erdős–Taylor conjecture. Period. Math. Hungar. 50 223–245. · Zbl 1098.60045
[11] Shi, Z. and Toth, B. (2000). Favourite sites of SRW. Period. Math. Hungar. 41 237–249. · Zbl 1001.60081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.