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GUE, GOE, hermitian/symmetric wigner matrices, ... already known with some mild assumptions. but is there any this kind of results without independent entries condition. thanks a lot.

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I would have thought that even a little searching on this topic would have brought up some examples. For instance, hermitian matrices where the joint density function of the entries is invariant under conjugation by unitaries - doesn't this get discussed in Mehta's book and other sources? –  Yemon Choi Jul 3 '10 at 23:47
as currently stated, the question reads a bit like a "fishing expedition". It might help people give useful answers if you can give some indication of what you've already seen and where you've tried looking. –  Yemon Choi Jul 3 '10 at 23:47

4 Answers 4

Another example is the adjacency matrix of a random regular graph. Here the entries are $0,1$ but the row sums and column sums must all be equal. For some properties of this matrix, see

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To add to this: The largest eigenvalue is governed by the Tracy Widom distribution (GOE). See –  Eric Naslund Jan 3 '12 at 11:15

Of course there is:

For example: this paper by Lubinsky describes universality for a special class of tridiginal matrices. For example the condition off-diagonals $\equiv 1$ and entries on the diagonal are in $\ell^1(\mathbb{Z})$ would suffice. So the matrices are $$ H_N = \begin{pmatrix} b_1 & 1 & \\\ 1 & b_2 & 1 & \\\ & 1 & b_3 &1 & \\\ & & \ddots & \ddots & \ddots \\\ & & & 1 & b_N \end{pmatrix} $$ with $\sum_{n=1}^{\infty} |b_n| < \infty$.

This can be further generalized see: Avila--Last--Simon. Of course all these results are for special tridiagonal matrices (Jacobi operators).

Last, there is also the work by Deift et al. See the book.

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the classic example is Dyson's circular ensemble of random unitary matrices (distributed uniformly with respect to the Haar measure); in the limit of large matrix size the correlation function of the eigenvalues tends to a universal limit.

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There are results on symmetric Gaussian matrices where the entries are dependent but the depedencies become weaker and weaker as the size of the matrix grows.

See e.g. and the references therein.

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