Limiting measures arising in the asymptotic theory of symmetric groups. I

A. M. Veršik   ; A. A. Šmidt

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 1, pp. 72-88

Résumé

We consider the joint distribution, with respect to the Haar measure, of a decreasing sequence of normalized lengths of cycles in a permutation, i. e. in an element of the symmetric group $S_n$ of degree $n$. We prove the existence of a limiting (as $n\to\infty$) distribution which is a measure in the space of non-negative series with unit sum. For this measure, it turns possible to find finite dimensional distributions and to study, in detail, its structure which is connected with some homogeneous Markov chain. This enables to obtain a large number of asymptotic formulas for invariant functionals on $S_n$, for example (main formula of Section 6), $$ \lim_i\lim_n\frac{1}{n!}|\{g\in S_n:(\ln(n_i(g)/n)+i)/\sqrt i\le b\}|=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^b e^{-x^2/2}\,dx, $$ where $n_i(g)$ is the length of the $i$-th, in size, cycle in the permutation $g\in S_n$.