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 $Z^2$ 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.