some $\theta\in[0,2\pi)$.

Since $$\bP\big(\S_{j}(\epsilon)\subseteq \S^{r}_{j}\big)=\bP\big(\S_{1}(\epsilon)\subseteq \S^{r}_{1}\big)$$ for all $j=1,...,k$, we only need to prove that $$\label{e1kshape}
 \lim_{r\to\infty}\bP\big(\S_{1}(\epsilon)\subseteq \S^{r}_{1}\big)=1\,.$$ To do so, for each $j=1,\dots,k$ let $\alpha^k_j:=\pi(j-1)/k$, $\vec{e}^k_j:=e^{2i\alpha^k_j}$ and $A_{r}:=\cap_{j=1}^{k} A^{j}_{r}$, where $$A_{r}^{j}:= \cap_{l\ne j}\big[H^{\alpha^k_j}(r\vec{e}^k_l,r\vec{e}^k_j)>0\big]\,.$$ Let $\alpha^{+\epsilon}_{k}:=\frac{\pi}{k}-\epsilon$ and $\alpha^{-\epsilon}_{k}:= (2\pi-\frac{\pi}{k})+\epsilon$ and set $$B_{r}(\epsilon):=\cap_{j=2,\dots,k}\big[H^{\alpha^{+\epsilon}_{k}}(r\vec{e}^k_j,r\vec{e}^k_1)>0\mbox{ and }H^{\alpha^{-\epsilon}_{k}}(r\vec{e}^k_j,r\vec{e}^k_1)>0\big]\,.$$ By Theorem \[tBuse-2\] $$\label{e2kshape}
\lim_{r\to\infty}\bP\big(A_{r}\cap B_r(\epsilon)\big)=1\,.$$ The connectivity of the regions $\B_j^r$ yields that, on $A_r\cap B_r(\epsilon)$, $\S_1(\epsilon)\subseteq \S_1^r$. Together with , this yields and the proof of Theorem \[t2\] is complete.

Auxiliary results {#pre}
=================

The probability space {#pre-ps}
---------------------

During the subsequent proofs we will consider the following construction of $(\Omega,\mathcal{F},\bP)$, the underline probability space of our FPP model. Let $\u_{0}=(0,0),\u_{2},\dots$ be a ordering of $\bZ^{2}$ and for each $k\geq 1$ let $$\B_{k}:=\u_{k}+[-1/2,1/2]^{2}\,.$$ Consider $$\calN=\{N_{k}\,:\, k\geq 1\},$$ a collection of i.i.d. Poisson random variables with intensity $1$; $$\calU_k=\{U_{k,l}\,:\, l\geq 1\},$$ a collection of independent random points in the plane so that $U_{k,l}$ has an uniform distribution in the square box $B_{k}$; $$\calT_k=\{\tau_{k,l}^{m,n}\,:\, l\geq 1,m\geq k,n\geq 1\mbox{ and }n>l\mbox{
  whenever }k=m\},$$ a collection of i.i.d. non negative random variables with common distribution $\bF$ (the passage time distribution). We also impose that all these collections are independent of each other.

To determine the vertex set $\calD_v=\calP$, 