Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions

  • Cox J
  • Durrett R
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Abstract

Let t(x, y) be the passage time from x to y in Z2 in a percolation process with passage time distribution F. If x ∈ R2 it is known that ∫ (1 - F(t))^4 dt < ∞) without any assumptions on F. The last two results describe the growth of the process in any fixed direction. We can also describe the asymptotic shape of At = y : t(0, y) ≤ t. Our results give necessary and sufficient conditions for t-1 At → x : φ(x) ≤ 1 in the sense of Richardson and show, without any assumptions on F, that the Lebesgue measure of t-1 AtΔx : φ(x) ≤ 1 → 0 almost surely. The last result can be applied to show that without any assumptions on F, the x-reach and point-to-line processes converge almost surely.

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Cox, J. T., & Durrett, R. (2007). Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions. The Annals of Probability, 9(4). https://doi.org/10.1214/aop/1176994364

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