The order of large random permutations with cycle weights

Abstract

The order $O_n(\sigma )$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k\ge 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdös and Turán who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We extend this result to the so-called \textit{generalized Ewens measure} in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furthermore, we provide precise large deviations estimates for random permutations with polynomial cycle weights.

41 pages, 5 figures

References in corpus (5)

• Random permutations with cycle weights
• Random permutation matrices under the generalized Ewens measure
• Mod-Gaussian convergence: new limit theorems in probability and number theory
• Distribution of Random Variables on the Symmetric Group
• Large cycles and a functional central limit theorem for generalized weighted random permutations