semi-infinite paths introduced by Newman [@n95].

\[t1\] For $k\geq 2$ let $\Omega_k$ be the event that, for the competing growth model with $k$-different species, there exists a finite subset $\Theta:=\{\theta_1,...,\theta_m\}$ of $[0,2\pi)$ such that every branch $\varphi$ of the competition interface is a $\theta(\varphi)$-path for some $\theta\in\Theta$. Under , $\bP\big(\Omega_k\big)=1$.

\[rand+strai\] In Section \[geo\] (part \[pr-rand+strai\]) we will give a sketch of the proof that for all $\alpha\in[0,2\pi)$ $$\bP\big(\theta=\alpha\mbox{ for some }\theta\in\Theta\big)=0\,,$$ and that if $\xi\in(3/4,1)$ then, almost surely, for all branch $\varphi=(\x_{n})_{n\geq 1}$ of the competition interface there is a constant $c>0$ such that $$ang(\x_{n},e^{i\theta(\varphi)})\leq c|\x_n|^{\xi-1}\mbox{ eventually }\,.$$

Let $\x_1(r)=(0,r),\dots,\x_k(r)$ be the vertices of a regular polygon with $k$ sides and radius $r$. For each $j=1,...,k$ define the projection of the random set $\B_{j}^r:=\B_{\x_j(r)}(\infty)$ onto $\S^1$, the set of unit vectors $|\x|=1$, by $$\S_{j,r}:=\{\x = e^{i\alpha}\in \S^{1}\,:\,\bL_{s\x}(\alpha)\subseteq \B_{j}^{r}\,\mbox{ for some }\,s>0\}\,,$$ where $\bL_\x(\alpha)$ denotes the line starting from $\x$ and with direction $e^{i\alpha}$. For each $\epsilon\in (0,\pi/k)$ and $j\in\{1,\dots,k\}$ define $$\S_{j}(\epsilon):=\{\x\in\S^1\,:\, ang(\x,\x_j(r))\leq \frac{\pi}{k}-\epsilon\}\,.$$

\[t2\] Let $k\geq 2$. Under , for all $\epsilon>0$ $$\lim_{n\to\infty}\bP\big(\S_{j}(\epsilon)\subseteq \S_{j,n}\,\emph{ for all $j=1,\dots,k$ }\big)=1\,.$$

Busemann type asymptotics and the competition interface
-------------------------------------------------------

To illustrate the approach we follow in this work to study the competition 