Late points for random walks in two dimensions

By Amir Dembo, Yuval Peres, Jay Rosen, Ofer Zeitouni. 2003

61 citations recorded by the source corpus.

Let τ n (x) denote the time of first visit of a point x on the lattice torus Z 2 n = 2 2 /nZ 2 by the simple random walk. The size of the set of a, n-late points £ n (α) = {x ∈ Z 2 n : Tn(x) ≥ α4 π (n log n) 2 } is approximately n 2(1-α) , for α ∈ (0,1) [£ n (α) is empty if a > 1 and n is large enough]. These sets have interesting clustering and fractal properties: we show that for β e (0, 1), a disc of radius n β centered at nonrandom x typically contains about n 2β(1-α/β2) points from £ n (α) (and is empty if β < √α), whereas choosing the center x of the disc uniformly in £ n (α) boosts the typical number of α, n-late points in it to n 2β(1-α) . We also estimate the typical number of pairs of a, n-late points within distance n β 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 n β 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 n β centered at a typical late point.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d3d0a3b1a621966a8396c5207564510e9ef6f8d9

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Citations in the corpus

Gumbel fluctuations for cover times in the discrete torus (2012, methodology): “for F = TN , where the entrance times of close vertices are far from being independent) we consider the set of (1− ρ)−late points (using the terminology from [10]), which is the set of vertices that are not yet hit at a 1 − ρ fraction of the typical time it takes to cover F , or more formally (3.”
Cover times for Brownian motion and random walks in two dimensions (2001, background): “See [14] for quantitative results on the uncovered set, based on the ideas of the present paper.”