the lattice and in the Euclidean FPP models.

Proof of the multi-type shape theorems {#multi}
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For each $j=1,...,k$, let $\S_{j}$ denote the set of unit vectors $e^{i\beta}$ such that $\bL_{se^{i\beta}}(\beta)\subseteq \B_{\x_j}(\infty)$ for some $s>0$ and let $$\S_{0}:=(\cup_{j=1}^{l}\S_{j})^{c}\,.$$ Let $$\D_n:=\{e^{i\beta}\,:\,\beta=2k\pi/2^{n}\mbox{ for some }1\leq k\leq 2^{n}\}\,$$ and $\D:=\cup_{n\geq 1}\D_n$. Consider the event $\cap_{\alpha\in\D}\Omega_0(\alpha)$ that for all $\alpha\in\D$ and $\v,\bar{\v}\in\calD_v$ there exists $H^{\alpha}(\v,\bar{\v})$, nonzero for $\v\neq\bar{\v}$, such that $$\lim_{{|\x|\to\infty}\atop{\x/|\x|\to e^{i\alpha}}}\big(T(\v,\x)-T(\bar{\v},\x)\big)=H^{\alpha}(\v,\bar{\v})\,.$$ By Theorem \[tBuse-1\], $\bP\big(\cap_{\alpha\in\D}\Omega_0(\alpha)\big)=1$.

We claim that, on this event, every branch of the competition interface is an $\theta$-path for some $\theta\in[0,2\pi)$. To see this, notice that if $e^{i\alpha}\in \S_{0}$ then for some $j_1 \ne j_2$, $\bL_{\0}(\alpha)$ intersects infinitely many times the region $\B_{\x_{j_1}}(\infty)$ and the region $\B_{\x_{j_2}}$. Thus $$\liminf_{s\to\infty}\big(T(\x_{j_1},se^{i\alpha})-T(\x_{j_2} ,se^{i\alpha})\big)\leq
0\leq\limsup_{s\to\infty}\big(T(\x_{j_1},se^{i\alpha})-T(\x_{j_2},se^{i\alpha})\big)\,,$$ which implies that $\D\cap\S_0=\emptyset$. Let $\C_k^n$ be the cone consisting of points $re^{i\beta}$ such that $r>0$ and $\beta\in(2\pi k/2^n,2\pi(k+1)/2^n)$. Now, if $\D\cap\S_0=\emptyset$ and $e^{i\beta}\in\D$ then every branch $\varphi$ of the competition interface can not intersect infinitely many times the line $\bL_{\0}(\beta)$. So, for each branch $\varphi$ of the competition interface we can find a sequence of cones $(\C^{n}_{k_{n}})_{n\geq1}$, with $n\to\infty$ and $\C^{n+1}_{k_{n+1}}\subseteq \C^{n}_{k_{n}}$, such that $\varphi$ is eventually contained in $C^{n}_{k_{n}}$. This implies that $\varphi$ must be a $\theta$-path for 