215 research outputs found
Late points for random walks in two dimensions
Let denote the time of first visit of a point on the
lattice torus by the simple random
walk. The size of the set of , -late points is approximately , for
[ is empty if and is large enough].
These sets have interesting clustering and fractal properties: we show that for
, a disc of radius centered at nonrandom
typically contains about points from
(and is empty if ), whereas
choosing the center of the disc uniformly in boosts
the typical number of -late points in it to .
We also estimate the typical number of pairs of , -late points
within distance of each other; this typical number can be
significantly smaller than the expected number of such pairs, calculated by
Brummelhuis and Hilhorst [Phys. A 176 (1991) 387--408]. On the other hand, our
results show that the number of ordered pairs of late points within distance
of each other is larger than what one might predict by multiplying
the total number of late points, by the number of late points in a disc of
radius centered at a typical late point.Comment: Published at http://dx.doi.org/10.1214/009117905000000387 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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