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 $(\frac{k-1}{g-1},\frac{a_k(S)}{g})$ where $1\le k\le g$ and $a_k(S)$ denotes the $k$th smallest nonzero element of $S$. We show that as $g\to \mathrm{\infty }$, 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.