any known universality results of random matrices with non-independent entries? - MathOverflow most recent 30 from http://mathoverflow.net2013-06-20T00:54:35Zhttp://mathoverflow.net/feeds/question/30468http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/30468/any-known-universality-results-of-random-matrices-with-non-independent-entriesany known universality results of random matrices with non-independent entries?joe2010-07-03T23:09:03Z2012-01-03T10:40:16Z
<p>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.</p>
http://mathoverflow.net/questions/30468/any-known-universality-results-of-random-matrices-with-non-independent-entries/30469#30469Answer by Helge for any known universality results of random matrices with non-independent entries?Helge2010-07-03T23:35:25Z2010-07-03T23:35:25Z<p>Of course there is:</p>
<p>For example: <a href="http://people.math.gatech.edu/~lubinsky/Research%20papers/UniversBulkJan07.pdf" rel="nofollow">this paper</a> 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 $\sum_{n=1}^{\infty} |b_n| < \infty$.</p>
<p>This can be further generalized see: Avila--Last--Simon. Of course all these results are for special tridiagonal matrices (Jacobi operators).</p>
<p>Last, there is also the work by Deift et al. See the <a href="http://books.google.com/books?id=SBR8yv0LkFgC&pg=PA237&dq=deift+universality&hl=en&ei=eMkvTJz7K4_csAag26y2Ag&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDsQ6AEwAw#v=onepage&q=deift%20universality&f=false" rel="nofollow">book</a>.</p>
http://mathoverflow.net/questions/30468/any-known-universality-results-of-random-matrices-with-non-independent-entries/48172#48172Answer by Carlo Beenakker for any known universality results of random matrices with non-independent entries?Carlo Beenakker2010-12-03T14:02:39Z2010-12-03T14:02:39Z<p>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.</p>
http://mathoverflow.net/questions/30468/any-known-universality-results-of-random-matrices-with-non-independent-entries/65164#65164Answer by vanvu for any known universality results of random matrices with non-independent entries?vanvu2011-05-16T18:39:21Z2011-05-16T18:39:21Z<p>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
<a href="http://arxiv.org/pdf/1011.6646.pdf" rel="nofollow">http://arxiv.org/pdf/1011.6646.pdf</a></p>
http://mathoverflow.net/questions/30468/any-known-universality-results-of-random-matrices-with-non-independent-entries/84755#84755Answer by Liviu Nicolaescu for any known universality results of random matrices with non-independent entries?Liviu Nicolaescu2012-01-02T17:09:35Z2012-01-03T10:40:16Z<p>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.</p>
<p>See e.g. <a href="http://arxiv.org/pdf/0707.2333.pdf" rel="nofollow">http://arxiv.org/pdf/0707.2333.pdf</a> and the references therein.</p>