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Báo cáo toán học: "Random Matrices, Magic Squares and Matching Polynomia"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Random Matrices, Magic Squares and Matching Polynomials. | Random Matrices Magic Squares and Matching Polynomials Persi Diaconis Alex Gamburd Departments of Mathematics and Statistics Department of Mathematics Stanford University Stanford CA 94305 Stanford University Stanford CA 94305 diaconis@math.stanford.edu agamburd@math.stanford.edu Submitted Jul 22 2003 Accepted Dec 23 2003 Published Jun 3 2004 MR Subject Classifications 05A15 05E05 05E10 05E35 11M06 15A52 60B11 60B15 Dedicated to Richard Stanley on the occasion of his 60th birthday Abstract Characteristic polynomials of random unitary matrices have been intensively studied in recent years by number theorists in connection with Riemann zetafunction and by theoretical physicists in connection with Quantum Chaos. In particular Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices magic squares . Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed. 1 Introduction Two noteworthy developments took place recently in Random Matrix Theory. One is the discovery and exploitation of the connections between eigenvalue statistics and the longest-increasing subsequence problem in enumerative combinatorics 1 4 5 47 59 another is the outburst of interest in characteristic polynomials of Random Matrices and associated global statistics particularly in connection with the moments of the Riemann zeta function and other L-functions 41 14 35 36 15 16 . The purpose of this paper is to point out some connections between the distribution of the coefficients of characteristic polynomials of random matrices and some classical problems in enumerative combinatorics. The second author .