of $T(\x,\y_n)-T(\0,\y_n)$ when $\y_n$ goes to infinity along a ray of angle $\alpha$ (Theorem \[tBuse-1\] and Theorem \[tBuse-2\]). We also study some roughening aspects of the one-dimensional boundary between the infections, namely the *competition interface*, which were pointed out by physicists in numerical simulations [@dd91; @sk95] (Remark \[rand+strai\]). We note that analogous problems in the context of last-passage percolation and totally asymmetric exclusion processes were treated by Ferrari and Pimentel [@fp04-1] and Ferrari, Martin and Pimentel [@fp04-2]. Deijfen, Häggström and Bagley [@dhb03] have also considered isotropic multi-type growth models in $\bR^d$ where the growth is driven by outbursts in the infected region.

Multi-type growth process
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Consider the random graph $\calD:=(\calD_v,\calD_e)$, named the *Delaunay triangulation*, constructed as follows. The vertex set $\calD_v\subseteq\bR^2$ is the set of points realized in a two-dimensional homogeneous Poisson point process with intensity $1$. To each vertex $\v$ corresponds an open and bounded polygonal region $\C_\v$ (the Voronoi tile at $\v$) consisting of the set of points of $\bR^2$ which are closer to $\v$ than to any other $\v'\in\calD_v$. The edge set $\calD_e$ consists of non oriented pairs $(\v,\v')$ such that $\C_\v$ and $\C_{\v'}$ share a one-dimensional edge (Figure \[f1\]). One can see that (with probability 