Energy scale below which ETH matrix elements become random-matrix-like

For a chaotic local Hamiltonian on $L$ sites and a local observable $O$, define $E_{\mathrm{RMT}}(L)$ as the largest energy difference $\omega$ such that off-diagonal elements $O_{mn}$ with $\left|E_m - E_n\right| < \omega$ are statistically invariant under unitary rotations, including the higher-order correlations of full ETH in its free-probability form. Determine $E_{\mathrm{RMT}}(L)$ for systems with and without diffusive conserved quantities; with diffusion the natural guess is $E_{\mathrm{RMT}} \sim \hbar D/L^{2}$. Answer: a scaling form established on at least one chaotic spin chain.

Computation of higher-order matrix-element correlations (free cumulants) in chaotic chains up to $L \sim 22-24$ by full diagonalization, with typicality methods for larger $L$, and a size scaling of the onset energy.