Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions

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.