Random Nilpotent Groups of Maximal Step

Let $G$ be a random torsion-free nilpotent group generated by two random words of length $\mathrm{\ell }$ in $U_n(\mathbb{Z})$. Letting $\mathrm{\ell }$ grow as a function of $n$, we analyze the step of $G$, which is bounded by the step of $U_n(\mathbb{Z})$. We prove a conjecture of Delp, Dymarz, and Schafer-Cohen, that the threshold function for full step is $\mathrm{\ell }=n^2$.