I'm currently reading the following reference on eigenstate thermalization and chaos in quantum mechanics: https://arxiv.org/abs/1509.06411
I'm confused by a derivation that I think is very important to understand properly; it's the subject of section 2.2.2, beginning on page 11. To summarize, the claim is the following: consider a Hermitian operator $\hat{O} = \sum_{i}O_{i}|i\rangle\langle i |$. Now assume we are given a random Hamiltonian $H$ with eigenvectors $|m\rangle, |n\rangle$. Use the eigenvectors of $H$ to look at matrix elements of $O$, e.g., $O_{mn} = \langle m|\hat{O}|n\rangle$. The text claims that (to first order in $\mathcal{D}$, the Hilbert space dimension)
$$\overline{\langle m|i\rangle\langle j|n\rangle} = \frac{1}{\mathcal{D}}\delta_{mn}\delta_{ij}$$
where the overline indicates "the average ... over random eigenkets $|m\rangle$ and $|n\rangle$". This result is not obvious to me at all. Perhaps my intuition is just failing because this is implicitly dealing with a large Hilbert space dimension. Clearly this depends on the observation that eigenvectors of random Hamiltonians are random orthogonal unit vectors, but I would like something closer to a proof to better understand this claim. I'm also unclear about what it means to average over random eigenkets (there is a comment in a later paragraph on how what we are really doing is summing over an ensemble of random Hamiltonians, but that confuses me even more frankly). Can anyone offer intuition or something closer to a proof for this statement?