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August, 1981 Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions
J. Theodore Cox, Richard Durrett
Ann. Probab. 9(4): 583-603 (August, 1981). DOI: 10.1214/aop/1176994364

Abstract

Let $t(x, y)$ be the passage time from $x$ to $y$ in $Z^2$ in a percolation process with passage time distribution $F$. If $x \in R^2$ it is known that $\int (1 - F(t))^4 dt < \infty$ is a necessary and sufficient condition for $t(0, nx)/n$ to converge to a limit in $L^1$ or almost surely. In this paper we will show that the convergence always occurs in probability (to a limit $\varphi(x) < \infty$) 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 $A_t = \{y : t(0, y) \leq t\}$. Our results give necessary and sufficient conditions for $t^{-1} A_t \rightarrow \{x : \varphi(x) \leq 1\}$ in the sense of Richardson and show, without any assumptions on $F$, that the Lebesgue measure of $t^{-1} A_t\Delta\{x : \varphi(x) \leq 1\} \rightarrow 0$ 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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J. Theodore Cox. Richard Durrett. "Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions." Ann. Probab. 9 (4) 583 - 603, August, 1981. https://doi.org/10.1214/aop/1176994364

Information

Published: August, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0462.60012
MathSciNet: MR624685
Digital Object Identifier: 10.1214/aop/1176994364

Subjects:
Primary: 60K99
Secondary: 60F15

Keywords: percolation processes , Richardson's model , Subadditive processes

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 4 • August, 1981
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