Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Jack deformations of Plancherel measures and traceless Gaussian random matrices. | Jack deformations of Plancherel measures and traceless Gaussian random matrices Sho Matsumoto Graduate School of Mathematics Nagoya University Furocho Chikusa-ku Nagoya 464-8602 Japan sho-matsumoto@ Submitted Oct 30 2008 Accepted Nov 28 2008 Published Dec 9 2008 Mathematics Subject Classihcation primary 60C05 secondary 05E10 Abstract We study random partitions A Al A2 . Ad of n whose length is not bigger than a Hxed number d. Suppose a random partition A is distributed according to the Jack measure which is a deformation of the Plancherel measure with a positive parameter a 0. We prove that for all a 0 in the limit as n 1 the joint distribution of scaled A1 . Ad converges to the joint distribution of some random variables from a traceless Gaussian P-ensemble with p 2 a. We also give a short proof of Regev s asymptotic theorem for the sum of p-powers of fx the number of standard tableaux of shape A. Key words Plancherel measure Jack measure random matrix random partition RSK correspondence 1 Introduction A random partition is studied as a discrete analogue of eigenvalues of a random matrix. The most natural and studied random partition is a partition distributed according to the Plancherel measure for the symmetric group. The Plancherel measure chooses a partition A of n with probability f A 2 A f-f- n where fx is the degree of the irreducible representation of the symmetric group n associated with A. A random partition A A1 A2 . chosen by the Plancherel measure is closely related to the Gaussian unitary ensemble GUE of random matrix theory. JSPS Research Fellow. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R149 1 The GUE matrix is a Hermitian matrix whose entries are independently distributed according to the normal distribution. The probability density function for the eigenvalues X1 xd of the d X d GUE matrix is proportional to e-2 U - x2d n Xj - Xj 1 i j d with p 2. In BOO J3 O1 see also BDJ it is proved that as n 1 the joint .
