any known universality results of random matrices with non-independent entries? 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.
 A: 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
A: 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.
A: 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.
A: 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.
