Discrete orthogonal polynomial ensembles and the Plancherel measure
The author considers orthogonal polynomial ensembles. The Meixner ensemble is related to a two-dimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zig-zag paths in random domino tilings of the Aztec diamond.NEWLINENEWLINENEWLINEThe author uses the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of \textit{C. A. Tracy} and \textit{H. Widom} [J. Stat. Phys. 92, No. 5-6, 809-835 (1998; Zbl 0942.60099)]. The author obtains the Plancherel measure on partitions, and using this the author proves a conjecture of \textit{J. Baik}, \textit{P. Deift} and \textit{K. Johansson} [Geom. Funct. Anal. 10, No. 4, 702-731 (2000; Zbl 0963.05133)] that under the Plancherel measure, the distributions of the lengths of the first \(k\) rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the \(k\) largest eigenvalues of a random matrix from the Gaussian unitary ensemble. In this problem a certain discrete kernel plays an important role.
scientific article; zbMATH DE number 1625953