The order of large random permutations with cycle weights

The order O-n (sigma) of a permutation sigma of n objects is the smallest integer k >= 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 Erdos and Turan 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.