Can you point me in the direction of any research done on the spectral theory (i.e. eigenvalues and eigenvectors) of real symmetric matrices with random (Gaussian or Levy) diagonal elements and fixed off-diagonal ones? Any links to papers, theorems or books will be appreciated.
There are two cases to consider depending on how your off-diagonal looks like:
If only two off-diagonal are non-zero, you are in the realm of random Schr\"odinger operators or random Jacobi operators. Then the special structure of the random variables is preserved up to a renormalization. In particular, the eigenvalues obey Poisson statistics in the limit as the matrix size goes to infinity.
The proof of this splits into two parts: First show that the eigenvectors decay exponentially in space. Then use this to decouple the eigenvalues.
A similar strategy should work as long as the number of non-zero off-diagonals remains small compared to the matrix size. But I am not sure if this has been proven. Probably not.
I have no idea what happens if many off-diagonals are non-zero. My best guess is that it's a mess.