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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
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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 http://arxiv.org/pdf/1011.6646.pdf

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To add to this: The largest eigenvalue is governed by the Tracy Widom distribution (GOE). See arxiv.org/PS_cache/arxiv/pdf/0903/0903.4295v5.pdf –  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. http://arxiv.org/pdf/0707.2333.pdf and the references therein.

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