The shape of a random numerical semigroup

We study statistical properties of random numerical semigroups of a given genus. We analyze the graph of a typical numerical semigroup, understood as a function from $\mathbb{N}$ to $\mathbb{N}$. If $S$ is a numerical semigroup of genus $g$, this leads us to consider the collection of points $\left(\frac{k-1}{g-1},\frac{a_k(S)}{g} \right)$ where $1 \le k \le g$ and $a_k(S)$ denotes the $k$th smallest nonzero element of $S$. We show that as $g \rightarrow \infty$, this set of points typically becomes closer to a union of two line segments. We prove analogous results for numerical semigroups ordered by Frobenius number.