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.
