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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.

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