Abstract. We study the model of random permutations of n objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size 1, 2, ..., b and a process (Z1, Z2, ..., Z b ) of independent Poisson random variables converges to 0 if and only if b = o(ℓ) where ℓ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend the Erdős-Turán Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.

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 Erdös and Turán who proved in 1965 that log On 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.Date: November 20, 2018. arXiv:1505.04547v1 [math.PR] 18 May 2015 Definition 1.1. Let Θ = (θ m ) m≥1 be given, with θ m ≥ 0 for every m ≥ 1. We then define for σ ∈ S n P n Θ [σ] := 1 h n n! n m=1 θ Cm m with h n = h n (Θ) a normalization constant and h 0 := 1. If n is clear from the context, we will just write P Θ instead of P n Θ .