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Late points for random walks in two dimensions. (English) Zbl 1100.60057
An issue of covering a finite lattice by a random walk [addressed before by M. Brummelhuis and H. Hilhorst, Physica A 176, 387-408 (1991)] is explored. A random walk on an \(n\times n\) square lattice with periodic boundary conditions is assumed to run until the cover time, when every point of the lattice has been visited. The focus is on the set of uncovered points, shortly before the ultimate coverage. These are called late points. In two dimensions, the set of such points is known to exhibit scaling properties typical for fractal structures, a property which is not present in higher dimensions. A quantitative description of the pertinent multifractal sets is given. Arguments in the paper rely on a direct analysis of the random walk, rather than on a strong approximation in terms of the Brownian motion.

MSC:
60K40 Other physical applications of random processes
60G50 Sums of independent random variables; random walks
28A80 Fractals
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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[1] Brummelhuis, M. and Hilhorst, H. (1991). Covering of a finite lattice by a random walk. Phys. A 176 387–408.
[2] Chung, K. L. (1974). A Course in Probability Theory , 2nd ed. Academic Press, New York. · Zbl 0345.60003
[3] 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
[4] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2004). Cover times for Brownian motion and random walks in two dimensions. Ann. Math. 160 433–464. · Zbl 1068.60018
[5] Fitzsimmons, P. and Pitman, J. (1999). Kac’s moment formula for additive functionals of a Markov process. Stochastic Process. Appl. 79 117–134. · Zbl 0962.60067
[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. · Zbl 1228.60004
[8] Lawler, G. (1993). On the covering time of a disc by a random walk in two dimensions. In Seminar in Stochastic Processes 1992 189–208. Birkhäuser, Boston. · Zbl 0789.60019
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