Gaussian Limit for the Plancherelmeasure

We show in this paper that the Plancherel measure of the symmetric group S n converges weakly (as n ! 1) to a Gaussian random process in the innnite dimensional vector space. With respect to the law of large numbers for this group, found earlier in 1,2] this assertion is analogous to the central limit theorem. The idea of such a result was mentioned in 3]. In the course of preparation of the article I had numerous valuable conversations with A.Vershik and G.Olshanski. In particular, it was Olshanski who suggested to use Lemma 2.3 below while discussing the preliminary version of the paper. 1. Main results. Let S n be the symmetric group of degree n and b S n { the set (of equivalence classes) of its irreducible representations. The character of a representation 2 b S n will be denoted as. If = (1 r 1 2 r 2 : : :) is a partition of n, we consider the corresponding class of conjugate permutations in S n with r k cycles of length k for k = 1; 2; : : : , and denote by the value of the character on this class. Let dim = (1 n) be the dimension of the representation 2 b S n. The weights M n () = dim 2 =n! determine the probability distribution M n on the set b S n called the Plancherel measure of the group S n. Deene the random variable ' k () = n k=2 (1.1) as the (normalized) value of characters of representations on the conjugacy class of permutations with a single non-trivial cycle of length k. The central result of the paper is the following Theorem 1. For any x 2 ; x 3 ; : : : ; x m 2 R there exists the limit lim n!1 M n f 2 b S n : ' k () < x k ; 2 k mg = m Y k=2 1 p 2k Z x k ?1 exp(? y 2 k 2k) dy k : (1.2) In other words, the functionals ' k () are asymptotically independent and have the Gaussian limiting distributions with zero mean value and with the variance k. Let us indicate a few alternative statements of the Theorem 1 dealing with other natural choices of main functionals. We shall use the standard identiication of the set …