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2015 The order of large random permutations with cycle weights
Julia Storm, Dirk Zeindler
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Electron. J. Probab. 20: 1-34 (2015). DOI: 10.1214/EJP.v20-4331


The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ 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 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.


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Julia Storm. Dirk Zeindler. "The order of large random permutations with cycle weights." Electron. J. Probab. 20 1 - 34, 2015.


Received: 27 May 2015; Accepted: 30 November 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1330.60019
MathSciNet: MR3433459
Digital Object Identifier: 10.1214/EJP.v20-4331

Primary: 60C05
Secondary: 60B15, 60F17


Vol.20 • 2015
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