Discrete orthogonal polynomial ensembles and the Plancherel measure. (English) Zbl 0984.15020

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.
The author uses the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of C. A. Tracy and 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 J. Baik, P. Deift and 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.


15B52 Random matrices (algebraic aspects)
60D05 Geometric probability and stochastic geometry
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