Concentration of functions of random unitary matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:33:11Z http://mathoverflow.net/feeds/question/97444 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97444/concentration-of-functions-of-random-unitary-matrices Concentration of functions of random unitary matrices Michal Kotowski 2012-05-19T23:23:47Z 2012-05-19T23:23:47Z <p>Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials $p(U,V)$ in entries of $U$ and $V$? More specifically, I am interested in polynomials of the form:</p> <p>$\sum U_{ij}V_{i'j'} X_{ii'}Y_{jj'}$</p> <p>or</p> <p>$|\sum U_{ij}V_{i'j'} X_{ii'}Y_{jj'}|^2$</p> <p>where $X$ and $Y$ are some arbitrary matrices and the sum is over all indices. For matrices with i.i.d. Gaussian entries there are well-known concentration bounds for this kind of expressions, I would like to know if there is anything similar for unitary matrices (for this case or for $U=V$).</p>