## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 4 (1981), 583-603.

### Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions

J. Theodore Cox and Richard Durrett

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 4 (1981), 583-603.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994364

**Digital Object Identifier**

doi:10.1214/aop/1176994364

**Mathematical Reviews number (MathSciNet)**

MR624685

**Zentralblatt MATH identifier**

0462.60012

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K99: None of the above, but in this section

Secondary: 60F15: Strong theorems

**Keywords**

Percolation processes Richardson's model subadditive processes

#### Citation

Cox, J. Theodore; Durrett, Richard. Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions. Ann. Probab. 9 (1981), no. 4, 583--603. doi:10.1214/aop/1176994364. https://projecteuclid.org/euclid.aop/1176994364