The shape of a random numerical semigroup

TL;DR

This paper investigates the asymptotic geometric structure of random numerical semigroups of large genus, revealing they tend to resemble two line segments as genus increases.

Contribution

It introduces a novel geometric perspective on the structure of random numerical semigroups and establishes their convergence to two line segments in the limit.

Findings

Points associated with semigroups approach two line segments as genus grows

The results hold for semigroups ordered by genus and Frobenius number

Provides a new geometric understanding of the typical shape of large semigroups

Abstract

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.