Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights

Abstract

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 of large random permutations with cycle weights

“…Recently, we studied the order of weighted permutations for polynomial parameters θ m = m γ , γ > 0, see [22]. We proved that the cycle counts of the cycles of length smaller than a typical cycle in this model can be decoupled into independent Poisson random variables.…”

“…[22],Lemma 4.6). For θ m = m γ with 0 < γ < 1 the following holds as n → ∞: P Θ ∆ n ≥ log(n) log log(n) → 0.…”

“…Recently, we proved in [22] that the cycle counts of the small cycles of length of order o(n 1 1+γ ) can be approximated by independent Poisson random variables. Using this result, we proved the Erdös-Turán law for this setting, see [22,Theorem 4.3].…”

“…The method we are applying to get our results is the saddle-point method. We will not repeat all details about this method here and refer the reader to Section 2.3 and Section 4.1 in [22]. 4.1.…”

“…and an appropriate method to investigate the behavior ofĝ Θ is the saddlepoint method. In [22, Lemma 4.1] we show thatĝ Θ is log-admissible (see Definition 2.8 in [22]). This gives us the asymptotic behavior of h n :…”