Asymptotics of Plancherel measures for symmetric groups

Alexei Borodin, Andrei Okounkov, Grigori Olshanski

Published 1999-05-05, updated 1999-09-11Version 2

We consider the asymptotics of the Plancherel measures on partitions of $n$ as $n$ goes to infinity. We prove that the local structure of a Plancherel typical partition (which we identify with a Young diagram) in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel. On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers math.CO/9810105 and math.CO/9901118 and from the combinatorial approach proposed by Okounkov in math.CO/9903176. Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures involving a new kernel on the 1-dimensional lattice. This kernel is expressed in terms of Bessel functions and we obtain it as a degeneration of the hypergeometric kernel from the paper math.RT/9904010 by Borodin and Olshanski. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.