I'm looking for possible resources with results related to the theory of random matrices (this field is pretty far from my comfort zone, so I am not even sure where to start looking).
In a problem I am working on, I have a random matrix $A$. I can consider the entries of its eigenvectors as random variables as well. I am interested in the expected value of these entries and in the correlation between them. Is anyone familiar with resources where such problems are treated, so that I can look for relevant information?
I am well aware that this problem is very general (I'm only looking for a starting point), so I'll give some more specifics (although anything would help):
My matrix is of the form $A=Oe^{iD}$, where $O$ is some fixed orthogonal matrix, and $D$ is a diagonal matrix who's diagonal entries are uniformly distributed on the $n$ dimensional cube $[0,2\pi]^n$.
I am aware that the entries of the eigenvectors are not 'well defined' random variables in the sense that there are infinite different choices for eigenvectors. For simplicity, assume that all eigenvalues of $A$ are simple (this is generically the case) and that I fix some 'gauge' which determines the eigenvector associated with each matrix $D$, in a way which varies smoothly across the given cube.
Numerically, I see that the expected value of the entries is $0$, and so is the correlation between different entries, and my goal is to prove it.
If you have any further questions about details of the problem, feel free to ask. Any help would be appreciated, thanks in advance.