Asymptotics of Plancherel measures for symmetric groups

Borodin, Alexei¹ ; Okounkov, Andrei^2, 3 ; Olshanski, Grigori⁴

¹ Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104–6395 and Dobrushin Mathematics Laboratory, Institute for Problems of Information Transmission, Bolshoy Karetny 19, 101447, Moscow, Russia
² University of Chicago, Department of Mathematics, 5734 University Ave., Chicago, Illinois 60637
³ Department of Mathematics, University of California at Berkeley, Evans Hall, Berkeley, California 94720-3840
⁴ Dobrushin Mathematics Laboratory, Institute for Problems of Information Transmission, Bolshoy Karetny 19, 101447, Moscow, Russia

Journal of the American Mathematical Society, Tome 13 (2000) no. 3, pp. 481-515

Résumé

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 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 and from the combinatorial proof given by the second author. Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures in terms of a new kernel involving Bessel functions. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.

Détail

DOI : 10.1090/S0894-0347-00-00337-4

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