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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.

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    $\begingroup$ Zeroes of Riemann Zeta and their relation to random matrices should provide a good deterministic choice...... $\endgroup$
    – Suvrit
    Aug 21, 2020 at 17:51
  • $\begingroup$ 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? $\endgroup$ Aug 21, 2020 at 18:05
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    $\begingroup$ These papers may be of interest: arxiv.org/abs/1701.05544 arxiv.org/abs/1702.04086 $\endgroup$ Aug 22, 2020 at 0:23

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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.

Spacing distributions are a different matter, but you did not ask about those....

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    $\begingroup$ I did ask about the level spacing distribution! (I've updated the question to make this clearer). Is there an an example for that case? But thank you for this partial answer. $\endgroup$ Aug 22, 2020 at 8:33
  • $\begingroup$ Lets not argue about what you asked originally. Wigner surmise is not the true distribution of spacings, in a precise mathematical sense, so if you go in the direction of spacings, you should be a bit more explicit as to what you look for. Are you trying to get the sine process? Let me mention also, following Suvrit, that for the function field case, the relation between the zeros of zeta function and RMT is a theorem, due to Katz-Sarnak. $\endgroup$ Aug 22, 2020 at 12:16
  • $\begingroup$ Thanks! Could you provide a reference explaining why Wigner surmise is not precisely the distribution of spacings? Supposing I replace my requirement for wigner surmise with the requirement that level correlations exhibit sine kernel behaviour in the appropriate scaling limit. Is there then an answer to my question? $\endgroup$ Aug 22, 2020 at 13:05
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    $\begingroup$ 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. $\endgroup$ Aug 22, 2020 at 13:20

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