Expected values of traces of products of random matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:51:34Z http://mathoverflow.net/feeds/question/97001 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97001/expected-values-of-traces-of-products-of-random-matrices Expected values of traces of products of random matrices Marcin Kotowski 2012-05-15T14:29:23Z 2012-09-06T04:25:41Z <p>Suppose I want to compute a quantity of the type: </p> <p>$\mathbb{E}\mathrm{tr}(AUBU^{\ast})$ </p> <p>where averaging is over Haar measure on the unitary group $\mathcal{U}(n)$ (one can of course consider higher order polynomials or other matrix ensembles etc.) and $A$, $B$ are some fixed matrices. Is there any standard technique for computing such averages? I'd guess people in random matrix theory or free probability compute such traces all the time, but I've been unable to find a reference. If it makes matters easier, I'm really interested in computing something for random projections (e.g. something of the form $\mathbb{E}\mathrm{tr}(APBP)$, where $P$ is a projection onto a random subspace), which of course reduces to computation of polynomials in $U$.</p> http://mathoverflow.net/questions/97001/expected-values-of-traces-of-products-of-random-matrices/98524#98524 Answer by Marcin Kotowski for Expected values of traces of products of random matrices Marcin Kotowski 2012-05-31T23:21:23Z 2012-05-31T23:21:23Z <p>Answering my own question, there is a closed formula for such traces, given in: <a href="http://arxiv.org/abs/math-ph/0402073" rel="nofollow">http://arxiv.org/abs/math-ph/0402073</a> (the formula involves representation theory of $S_n$ and gets ugly as $n$ gets bigger, but can be written down explicitly at least for small $n$)</p> http://mathoverflow.net/questions/97001/expected-values-of-traces-of-products-of-random-matrices/106474#106474 Answer by Jiahao Chen for Expected values of traces of products of random matrices Jiahao Chen 2012-09-06T04:25:41Z 2012-09-06T04:25:41Z <p>For the unitary group, the first paper I am aware of to do these sorts of averages is:</p> <p><a href="http://link.aip.org/link/JMAPAQ/v21/i12/p2695/s1" rel="nofollow">http://link.aip.org/link/JMAPAQ/v21/i12/p2695/s1</a></p> <p>An early paper of Collins' in 2003 expresses such averages in terms of Weingarten functions, which are usually expressed as character expansions over $U(N)$, $O(N)$ or $Sp(N)$.</p> <p><a href="http://arxiv.org/abs/math-ph/0205010" rel="nofollow">http://arxiv.org/abs/math-ph/0205010</a></p> <p>Some early papers calculating these character expansions are:</p> <p><a href="http://jmp.aip.org/resource/1/jmapaq/v25/i6/p2028_s1" rel="nofollow">http://jmp.aip.org/resource/1/jmapaq/v25/i6/p2028_s1</a></p> <p><a href="http://jmp.aip.org/resource/1/jmapaq/v43/i1/p604_s1" rel="nofollow">http://jmp.aip.org/resource/1/jmapaq/v43/i1/p604_s1</a></p>