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The order of large random permutations with cycle weights. (English) Zbl 1330.60019
Summary: 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 P. Erdős and P. Turán who proved in [Z. Wahrscheinlichkeitstheor. Verw. Geb. 4, 175–186 (1965; Zbl 0137.25602)] that \(\log O_n\) satisfies a central limit theorem. We extend this result to the so-called generalized Ewens measure in a previous paper [the authors, “Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights”, Ann. Inst. Henri Poincaré (to appear), arXiv:1410.5406]. 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.

60C05 Combinatorial probability
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F17 Functional limit theorems; invariance principles
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