Statistics of Erdős-Rényi random numerical semigroups

For $p>0$ a small parameter, let $\mathcal{A}\subseteq {\mathbb{Z}}_{>0}$ be a random subset where each positive integer is included independently with probability $p$. We show that, with high probability (as $p\to 0$), the numerical semigroup $⟨\mathcal{A}⟩:=\{a_1+\cdots +a_k:k\ge 0,a_1,\dots ,a_k\in \mathcal{A}\}$ generated by $\mathcal{A}$ has Frobenius number and genus of size $\asymp p^{-1}(\log p^{-1}{)}^2$ and embedding dimension of size $\asymp (\log p^{-1}{)}^2$. This resolves an open problem of Bogart and the second author.