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Asymptotic orders of expected Frobenius number and embedding dimension in the ER-type model

Establish that, in the Erdős–Rényi-type random numerical semigroup model (p) where each positive integer is independently included as a generator with probability p, the expected Frobenius number E[F((p))] is Θ((1/p) log(1/p)) and the expected embedding dimension E[e((p))] is Θ(log(1/p)) as p → 0.

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Background

After improving upper bounds for the expected Frobenius number and embedding dimension in the ER-type model, the authors compare them to known lower bounds and to experimental data. The experiments suggest the true growth rates involve exactly one factor of log(1/p), lying strictly between the previous lower bounds and the new upper bounds.

Motivated by this empirical evidence, the authors formulate a conjecture specifying the asymptotic orders for both the expected Frobenius number and the expected embedding dimension.

References

However, extensive experiments , in which 1000 Erd\H{o}s-R\ enyi semigroups were generated for each of fifteen values of $p$, suggest the following conjecture. (1) The expected Frobenius number $[\F((p))]$ is of order $\frac{1}{p} \log \left(\frac{1}{p}\right)$. (2) The expected embedding dimension $[\e((p))]$ is of order $\log \left(\frac{1}{p}\right)$.

— Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups (2411.13767 - Bogart et al., 21 Nov 2024) in Section 5: Experiments, Conclusions, and Future Work