Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Moments of characteristic polynomials enumerate two-rowed lexicographic arrays | Moments of characteristic polynomials enumerate two-rowed lexicographic arrays E. Strahov Department of Mathematical Sciences Brunel University Uxbridge UB8 3PH United Kingdom Submitted Nov 21 2001 Accepted May 14 2003 Published May 29 2003 MR Subject Classifications 15A52 20G05 11M06 Abstract A combinatorial interpretation is provided for the moments of characteristic polynomials of random unitary matrices. This leads to a rather unexpected consequence of the Keating and Snaith conjecture the moments of IC 1 2 it turn out to be connected with some increasing subsequence problems such as the last passage percolation problem . 1 Introduction Keating and Snaith 1 have proposed to model the limiting distribution of the Riemann zeros using the characteristic polynomials of unitary random matrices U Z U 0 det I - e idU . 1 In particular they deal with a conjecture for moments of I 1 2 it l which states that the limit Im ri O Tfm2 f l 1 2 it 2mdt 2 T log T m 0 exists and is equal to a product of two factors f m and a m i. e. Im a m f m . 3 The first factor a m is the zeta-function specific part a m Y1 h m2 r k m k a m ụ 1 - p 2 a r mj p 4 THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R24 1 where the product is taken over prime numbers p. As for the second factor f m Keating and Snaith 1 have hypothesized that it is the random matrix universal part and may be represented as f m lim N-m z U ỡ 2m u N . 5 N Here the average is over the Circular Unitary Ensemble CUE of random unitary matrices N X N. In this paper I provide a combinatorial interpretation for the moments of characteristic polynomials z U Q 2m u N . I then relate these moments with two-rowed lexicographic arrays which are generalizations of permutations and words in combinatorics. For basic information about permutations words and lexicographic arrays see for example a book by Fulton 2 . The combinatorial interpretation of moments of the characteristic polynomials enables an .
