\B_\0(t)\subseteq (1+\epsilon )t
\D(1/\mu)\mbox{ eventually }\big)=1 \,,$$ where $\D(r):=\{\x\in\bR^2 \,:\, |\x|\leq r\}$ and $\0:=(0,0)$.

When $k\geq2$ the process $\big\{(\B_{\x_1}(t),...,\B_{\x_k}(t))\,:\,t\geq 0\big\}$ is a model for competing growth on the plane where each point $\x\in\bR^2$ is acquired by the specie $j\in\{1,\dots,k\}$ which first arrives there. The competition interface $\psi$ is the one-dimensional boundary between the species when $t=\infty$. This interface can be seen as a finite union of polygonal curves determined by edges in $\calV$ (the Voronoi tessellation) which are shared by tiles in different species. A branch of the competition interface is a self-avoiding path $\varphi=(\x_n)_{n\geq 1}$ in $\calV$ such that $\{\x_n\,:\,n\geq 1\}\subseteq\psi$.

For each $\alpha\in[0,2\pi)$ we say that a self-avoiding path $(\x_{n})_{n\geq 1}$, with vertices in $\bR^2$ and such that $|\x_n|\to\infty$ when $n\to\infty$, is a $\alpha$-path if $$\lim_{n\to\infty}\frac{\x_{n}}{|\x_{n}|}=e^{i\alpha}:=(\cos\alpha,\sin\alpha)\,.$$ In this case we also say that $(\x_n)_\bN$ has the asymptotic orientation $e^{i\alpha}$. This is equivalent to $$\lim_{n\to\infty}ang(\x_{n},e^{i\alpha})=0\,,$$ where $ang(\x,\y)$ denotes the angle in $[0,\pi]$ between the points $\x,\y\in\bR^2$. Thus, a sufficient condition for a path $(\x_{n})_{n\geq 1}$ to be a $\alpha$-path for some $\alpha\in[0,2\pi)$ is, for some fixed $\delta\in(0,1)$ and some constant $c>0$, for sufficiently large $n$ $$ang(\x_{n},\x_m)\leq |\x_n|^{-\delta}\,\mbox{ whenever }m > n\,,$$ which is the so called $\delta$-straightness property for 