Long cycle of random permutations with polynomially growing cycle weights

Abstract

We study random permutations of n objects with respect to multiplicative measures with polynomial growing cycle weights. We determine in this paper the asymptotic behavior of the long cycles under this measure and also prove that the cumulative cycle numbers converge in the region of the long cycles to a Poisson process. KEYWORDScycle counts, long cycles, Poisson process, random permutations, saddle point method INTRODUCTIONLet n be the symmetric group of all permutations on elements {1, … , n}. For any permutation ∈ n , denote by C m = C m ( ) the cycle counts, that is, the number of cycles of length m = 1, … , n in the cycle decomposition of ; clearly(1.1) Random permutationsClassical probability measures studied on n are the uniform measure and the Ewens measure. The uniform measure is well studied and has a long history (see e.g., the first chapter of [1] for a detailed account with references). The Ewens measure originally appeared in population genetics, see [11], but has also various applications through its connection with Kingman's coalescent process, see [13].Classical results about uniform and Ewens random permutations include convergence of joint cycle counts towards independent Poisson random variables in total variation distance [2] and a central limit theorem for cumulative cycle counts [7]. Furthermore, the longest cycles have order of magnitude n and it was established by Kingman ([14]) and by Vershik and Shimdt ([20]) that the vector of renormalized and ordered length of the cycles converges in law to a Poisson-Dirichlet distribution.In this paper, we study random permutations with respect to the probability measure 726