tailieunhanh - Báo cáo toán học: "Truncations of Random Unitary Matrices and Young Tableaux"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Truncations of Random Unitary Matrices and Young Tableaux. | Truncations of Random Unitary Matrices and Young Tableaux J. Novak Department of Mathematics and Statistics Queen s University Kingston Canada jnovak@ Submitted Aug 6 2006 Accepted Jan 30 2007 Published Feb 7 2007 Mathematics Subject Classification 05E10 Abstract Let U be a matrix chosen randomly with respect to Haar measure from the unitary group U d . For any k d and any k X k submatrix Uk of U we express the average value of I Tr Uk 2n as a sum over partitions of n with at most k rows whose terms count certain standard and semistandard Young tableaux. We combine our formula with a variant of the Colour-Flavour Transformation of lattice gauge theory to give a combinatorial expansion of an interesting family of unitary matrix integrals. In addition we give a simple combinatorial derivation of the moments of a single entry of a random unitary matrix and hence deduce that the rescaled entries converge in moments to standard complex Gaussians. Our main tool is the Weingarten function for the unitary group. 1 Introduction and Statement of the Main Theorem Consider the unitary group U d as a probability space under normalized Haar measure dU. Given a random variable X U d C its expected value is defined to be Eu d X L. Ju d XdU. When studying a random variable X one often wishes to know its moments Eu d X mXn since in many situations the moments of X uniquely determine its distribution. X is called a polynomial random variable if it is polynomial in the entries of U . if there is a polynomial f 2 C xn . xdd such that X U f uu . Udd THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R21 1 for all U 2 U d . If X is a polynomial random variable then Eu d XmX 0 if m n see 1 so knowledge of the moments of X reduces to knowledge of the quantities Eu d X 2n . It has recently been shown that certain polynomial random variables on the unitary group encode interesting combinatorial information via their moments. For a random matrix U 2 U d write its characteristic