any known universality results of random matrices with non-independent entries? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:54:35Z http://mathoverflow.net/feeds/question/30468 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30468/any-known-universality-results-of-random-matrices-with-non-independent-entries any known universality results of random matrices with non-independent entries? joe 2010-07-03T23:09:03Z 2012-01-03T10:40:16Z <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#30469 Answer by Helge for any known universality results of random matrices with non-independent entries? Helge 2010-07-03T23:35:25Z 2010-07-03T23:35:25Z <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 $$H_N = \begin{pmatrix} b_1 &amp; 1 &amp; \\ 1 &amp; b_2 &amp; 1 &amp; \\ &amp; 1 &amp; b_3 &amp;1 &amp; \\ &amp; &amp; \ddots &amp; \ddots &amp; \ddots \\ &amp; &amp; &amp; 1 &amp; b_N \end{pmatrix}$$ with $\sum_{n=1}^{\infty} |b_n| &lt; \infty$.</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&amp;pg=PA237&amp;dq=deift+universality&amp;hl=en&amp;ei=eMkvTJz7K4_csAag26y2Ag&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=4&amp;ved=0CDsQ6AEwAw#v=onepage&amp;q=deift%20universality&amp;f=false" rel="nofollow">book</a>.</p> http://mathoverflow.net/questions/30468/any-known-universality-results-of-random-matrices-with-non-independent-entries/48172#48172 Answer by Carlo Beenakker for any known universality results of random matrices with non-independent entries? Carlo Beenakker 2010-12-03T14:02:39Z 2010-12-03T14:02:39Z <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#65164 Answer by vanvu for any known universality results of random matrices with non-independent entries? vanvu 2011-05-16T18:39:21Z 2011-05-16T18:39:21Z <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#84755 Answer by Liviu Nicolaescu for any known universality results of random matrices with non-independent entries? Liviu Nicolaescu 2012-01-02T17:09:35Z 2012-01-03T10:40:16Z <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>