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Thick Points for Intersections of Planar Sample Paths

Amir Dembo, Yuval Peres, Jay Rosen and Ofer Zeitouni
Transactions of the American Mathematical Society
Vol. 354, No. 12 (Dec., 2002), pp. 4969-5003
Stable URL: http://www.jstor.org/stable/3072975
Page Count: 35
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Thick Points for Intersections of Planar Sample Paths
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Abstract

Let LXn(x) denote the number of visits to x ∈ Z2 of the simple planar random walk X, up to step n. Let X' be another simple planar random walk independent of X. We show that for any 0 < b < l/(2π), there are n1-2π b+o(1) points x ∈ Z2 for which LXn(x)LX'n (x) ≥ b2(log n)4. This is the discrete counterpart of our main result, that for any a < 1, the Hausdorff dimension of the set of thick intersection points x for which lim supr→ 0 I(x, r)/(r2| log r|4) = a2, is almost surely 2 - 2a. Here I(x, r) is the projected intersection local time measure of the disc of radius r centered at x for two independent planar Brownian motions run until time 1. The proofs rely on a "multi-scale refinement" of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius r centered at x by x + rK for general sets K.

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