Deterministic matrices with random matrix properties

A matrix chosen randomly from the Gaussian Orthogonal Ensemble of $$n\times n$$ matrices has an empirical eigenvalue distribution which (suitably coarse-grained) follows a Wigner semi-circle law (as $$n\rightarrow \infty$$). Similar statements are believed to hold for, e.g., empirical level spacing distribution functions (Wigner surmise). Thus a particular (large) random matrix exhibits the average properties of its home ensemble.

My question: Is there an explicitly known sequence of $$n\times n$$ real symmetric matrices $$\{ A_n \}^{\infty}_{n=1}$$ for which one can prove $$A_n$$ develops random matrix spectral qualities as $$n\rightarrow \infty$$. For example, can you give an explicit construction of a sequence of such $$A_n$$ for which one can prove the empirical eigenvalue and level-spacing distributions approach the semi-circle law and Wigner surmise respectively?

By "explicitly known", I'd ideally want a deterministic formula for the matrix elements $$[A_n]_{ij}$$. So, for example, I'd be happy if you could prove the above spectral features emerge if you set $$[A_n]_{i\geq j}=(i\times j)^{\text{th}} \text{ digit of }\pi$$. As it turns out, a quick numerical experiment suggests that the eigenvalue gap distribution for this example converges to the GOE result. But I'm interested in a proof, and preferably a less cumbersome construction.

• Zeroes of Riemann Zeta and their relation to random matrices should provide a good deterministic choice...... Aug 21, 2020 at 17:51
• I'm aware that the pair correlations of eigenvalues of GOE matrices are conjectured to be the same as those between $\zeta$ zeros. How does one go from that conjectured connection to the explicit matrix I'm asking for? Aug 21, 2020 at 18:05
• These papers may be of interest: arxiv.org/abs/1701.05544 arxiv.org/abs/1702.04086 Aug 22, 2020 at 0:23

Actually, just to get the semicircle is not hard. Take an $$n$$-by-$$n$$ Jacobi matrix whose on diagonal entries are $$0$$ and $$i$$th entry on the off diagonal is $$\sqrt{i/n}$$. The limit ESD will be the semi-circle.
The reason this works is that what you would have constructed is the mean part of the Dumitriu-Edelman Jacobi model for G$$\beta$$E, which will have the same limit for the empirical density of state. See https://arxiv.org/abs/math-ph/0206043 or the journal publication for details.
• en.wikipedia.org/wiki/Wigner_surmise describes Wigner's surmise. The law there is not the one you get from the sine process (see Mehta), but is close numerically. About your question, I suspect you can cook it up by modifying, in the Jacobi construction, the the off-diagonal entries in a periodic way (scale $\sqrt{1/k}$), but I have not done the computation. Aug 22, 2020 at 13:20