$R^*$ be a copy of $R$ that is independent of $X^{(n)}$. Then, $$D^{(\alpha)}_{\mathsf{out},n}=\sum_{(\xi,r)\in X^{(n)}}|\xi|^\alpha1_{\BT_{R^*}(o)}(\xi),$$ denotes the sum of $\alpha$th powers of the lengths of all outgoing edges at the node $(o,R^*)$, where we put $|\xi|=\d_{\T_n}(\xi,o)$. Considering the limit $n\to\infty$, i.e., letting the size of the torus tend to infinity, it is intuitive (and will be shown rigorously in Theorem \[outEdgeThm\] below) that the random variables $\big\{D^{(\alpha)}_{\mathsf{out},n}\big\}_{n\ge1}$ converge in distribution to the random variable $$D^{(\alpha)}_{\mathsf{out}}=\sum_{(\xi,r)\in X}|\xi|^\alpha1_{B_{R^*}(o)}(\xi),$$ where $X$ denotes an independently $[0,\infty)$-marked homogeneous Poisson point process in $\R^d$ with intensity $1$. In addition to showing the convergence of $D^{(\alpha)}_{\mathsf{out},n}$ to $D^{(\alpha)}_{\mathsf{out}}$, we also investigate the behavior of the tail probabilities $p^{(\alpha)}_{\mathsf{out},t}=\P\big(D^{(\alpha)}_{\mathsf{out}}>t\big)$ as $t\to\infty$. In the following, $\kappa_d$ denotes the volume of the unit ball in $\R^d$.

\[outEdgeThm\] For every $\alpha>0$ the random variables $\big\{D^{(\alpha)}_{\mathsf{out},n}\big\}_{n\ge1}$ converge to the random variable $D^{(\alpha)}_{\mathsf{out}}$ in distribution. Moreover, $$\lim_{t\to\infty}t^{s/(\alpha+d)}p^{(\alpha)}_{\out,t}=(d\kappa_d/(\alpha+d))^{s/(\alpha+d)}\beta.$$

, the degree distribution of out-degrees admits asymptotically polynomial tails of order $s/d$. In contrast, when considering in-degrees an entirely different asymptotic behavior emerges. Indeed, in Theorem \[inEdgeThm\] below we show that for $s\le d$ the degree distribution is asymptotically degenerate, whereas for $s>d$ it is Poissonian (with finite mean). In order to make this precise, let 