As genus g tends to infinity, the scaled graph of a typical numerical semigroup of genus g converges to two line segments.
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A numerical semigroup is a collection of whole numbers that is closed under addition and misses only finitely many numbers. The genus g counts how many numbers are missing. The authors pick a random semigroup with exactly g missing numbers and look at its smallest g+1 elements. They plot each element's position after scaling the index by 1/g and the value by 1/g. In the limit of very large g, these plotted points line up along two straight segments instead of scattering randomly. The same two-segment shape appears when semigroups are chosen by fixing the largest missing number instead of the genus. The result is proved by combining counting arguments with concentration inequalities that show most semigroups behave like the average one.
Core claim
We show that as g → ∞, this set of points typically becomes closer to a union of two line segments.
Load-bearing premise
The uniform probability measure on the set of all numerical semigroups of fixed genus g is the correct model for a 'typical' semigroup, and the limit shape exists under this measure.
read the original 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.
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