i.e. $T(\0,\x)\leq t$. One may think of sites in $\x\in \B(t)$ as infected and those in $\B(t)^c$ as healthy, and that at time $0$ the origin $\0$ is infected by some type of disease. The process $\big\{\B(t)\,:\,t\geq 0\big\}$ is then a model for the growth of an infection.

An interesting aspect of the evolution of the infection, namely the *tree of infection*, is constructed as follows. First notice that, since the passage time distribution is continuous, for all $\x,\y\in\bZ^2$ there is (almost surely) an unique time-minimizing path (or geodesic) from $\x$ to $\y$, which we denote $\rho(\x,\y)$, such that $T(\x,\y)=t\big(\rho(\x,\y)\big)$. Thus $\rho(\x,\y)$ may be interpreted as the path through which the infection was transmitted from $\0$ to $\x$. With this picture in mind, the tree of infection $\Gamma$ is defined by the union of edges $\e\in\rho(\0,\x)$ over all $\x\in\bZ^2$. Newman [@n95] has shown that the number $K(\Gamma)$ of topological ends of $\Gamma$, i.e. the number of semi-infinite self-avoiding paths in $\Gamma$, is infinite provided an exponential moment condition on $\bF$ and a certain hypothesis concerning the uniformly bounded curvature of the asymptotic shape of $\B(t)$. In spite of the curvature hypothesis is plausible it has so far not been proved.

