Abstract
The order On(σ) of a permutation σ of n objects is the smallest integer k≥1 such that the k-th iterate of σ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdos and Turan who proved in 1965 that logOn satisfies a central limit theorem. We extend this result to the so-called 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.
Highlights
Denote by Sn the symmetric group, that is the group of permutations on n objects
We present large deviations estimates and a local limit theorem for log On which are, to our knowledge, new even for the uniform measure
We focus on random permutations with cycle weights as introduced in the recent works of Betz et al [3] and Ercolani and Ueltschi [7]
Summary
Denote by Sn the symmetric group, that is the group of permutations on n objects. For a permutation σ ∈ Sn the order On = On(σ) is defined as the smallest integer k such that the k-th iterate of σ is the identity. In this paper we study the random variable log On with respect to a weighted measure. Many properties of permutations considered with respect to this weighted measure have been examined for different classes of parameters, see for instance [3, 7, 11, 15, 16, 17, 18, 19]. We proved that the cycle counts of the cycles of length smaller than a typical cycle in this model can be decoupled into independent Poisson random variables Using this approximation, we extended the Erdös-Turán law (1.1) to this setting as well as a functional version of it.
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