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Hitting probabilities of random walks on

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  • Kesten, Harry

Abstract

Let S0, S1, ... be a simple (nearest neighbor) symmetric random walk on and HB(x,y) = P{S. visits B for the first time at yS0 = x}. If d = 2 we show that for any connected set B of diameter r, and any y [epsilon] B, one has lim sup HB(x, y) [less-than-or-equals, slant] C(2) r-1/2 · x --> [infinity] If d [greater-or-equal, slanted] 3 one has for any connected set B of cardinality n, lim sup HB(x, y) [less-than-or-equals, slant] C(d)n-1+2/d · x --> [infinity] These estimates can be used to give bounds on the maximal growth rate of diffusion limited aggregation, a fashionable growth model for various physical phenomena.

Suggested Citation

  • Kesten, Harry, 1987. "Hitting probabilities of random walks on," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 165-184.
    • Handle: RePEc:eee:spapps:v:25:y:1987:i::p:165-184
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    Cited by:

    1. Amir, Gideon & Benjamini, Itai & Gurel-Gurevich, Ori & Kozma, Gady, 2020. "Random walk in changing environment," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7463-7482.
    2. Procaccia, Eviatar B. & Zhang, Yuan, 2021. "On sets of zero stationary harmonic measure," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 236-252.
    3. Kôhei Uchiyama, 2016. "The Hitting Distribution of a Line Segment for Two-Dimensional Random Walks," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1661-1684, December.

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