interface assume that $k=2$. Consider the line $\bL_\0(\alpha)$ starting from the origin $\0$ and with direction $e^{i\alpha}$. Then we have three possibilities: i) either it intersets the competition interface infinitely many times; ii) or it is eventually contained in $\B_{\x_1}(\infty)$; iii) or it is eventually contained in $\B_{\x_2}(\infty)$. Notice that the former implies $$\liminf_{s\to\infty}\big(T(\x_{1},se^{i\alpha})-T(\x_{2} ,se^{i\alpha})\big)\leq
0\leq\limsup_{s\to\infty}\big(T(\x_{1},se^{i\alpha})-T(\x_{2},se^{i\alpha})\big)\,,$$ while the second implies $$\limsup_{s\to\infty}\big(T(\x_{1},se^{i\alpha})-T(\x_{2} ,se^{i\alpha})\big)\leq 0\,,$$ and the third implies $$0\leq \liminf_{s\to\infty}\big(T(\x_{1},se^{i\alpha})-T(\x_{2} ,se^{i\alpha})\big)\,.$$ It turns out that the above expressions resemble Busemann type asymptotics for $T$ (see Ballmann [@b95] for more details on this subject). Newman [@n95; @ln96] has shown for the lattice model that, under suitable assumptions on the curvature of the limit shape, $T(\x_1,\y_n)-T(\x_2,\y_n)$ attains eventually a nonzero value $H^{\alpha}(\x_1,\x_2)$, called the Busemann function. By following his method, and by taking profit of the isotropy in our model, we will show[^1]:

\[tBuse-1\] For $\alpha\in[0,2\pi)$ let $\Omega_0(\alpha)$ be the event that for all $\v,\bar{\v}\in\calD_v$, there exists $H^{\alpha}(\v,\bar{\v})$, nonzero for $\v\neq\bar{\v}$, such that $$\label{eBuse-1}
\lim_{{|\x|\to\infty}\atop{\x/|\x|\to e^{i\alpha}}}\big(T(\v,\x)-T(\bar{\v},\x)\big)=H^{\alpha}(\v,\bar{\v})\,.$$ Under , $\bP\big(\Omega_0(\alpha)\big)=1$.

For $\x ,\y\in\bR^2$ we set $H^\alpha(\x,\y):=H^\alpha\big(\v(\x),\v(\y)\big)$. It was conjectured by Howard and Newman [@hn01] that $$\lim_{n\to\infty}\frac{H^{\alpha}(n\vec{e}_1,\0)}{n}=-\mu(\bF)\cos\alpha\,,$$ where $\vec{e}_1:=(1,0)$. This observation is related to the asymptotic behavior of our multi-type growth model and the key 