Consider a class of real symmetric random matrices,$M_{n\times n}=(X_{i,j})_{n\times n},$ whose off-diagonal elements follow an exchangeable distribution, and the diagonal elements follow another exchangeable distribution and the diagonal part and off-diagonal part are independent. My question is what's the eigenvalue density of this type of random matrices? Is the eigenvalue density Universal?

I know that Chatterjee has a paper: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aop/1171377437

But my matrices are different from his consideration. I do have a specific example. If we choose the off-diagonal part follow Dirichlet distribution and the diagonal part follow another Dirichlet distribution, then the eigenvalue density is some Gamma distribution. To my opinion, this is actually trivial, for after we rescale the matrix, it's asymptotic matrix is an infinite dimensional identity matrix. What about other cases?