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.


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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