Permutation Matrices and the Moments of their Characteristic Polynomials

In this paper, we are interested in the moments of the characteristic polynomial $Z_n(x)$ of the $n\times n$ permutation matrices with respect to the uniform measure. We use a combinatorial argument to write down the generating function of \E{\prod_{k=1}^p Z_n^{s_k}(x_k)} for s_k\in\Nr. We show with this generating function that \lim_{n\rw\infty} \E{\prod_{k=1}^p Z_n^{s_k}(x_k)} exists for $\max_k|x_k|<1$ and calculate the growth rate for $p=2, |x_1|=|x_2|=1$, $x_1=\overline{x_2}$ and n\rw\infty. We also look at the case s_k\in\C. We use the Feller coupling to show that for each $|x|<1$ and s\in\C there exists a random variable $Z_\infty^s(x)$ such that $Z_n^s(x)\xrightarrow{d}Z_\infty^s(x)$ and \E{\prod_{k=1}^p Z_n^{s_k}(x_k)}\rw \E{\prod_{k=1}^p Z_\infty^{s_k}(x_k)} for $\max_k|x_k|<1$ and n\rw\infty.Comment: 24 pages, 1 Figur

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 $k \geq 1$ 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 Erd\"os and Tur\'an who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We extend this result to the so-called \textit{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.Comment: 41 pages, 5 figure

The limit shape of random permutations with polynomially growing cycle weights

In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set $\{1,\dots,n\}$ under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process.Comment: 36 pages, 3 figures. The paper was subject to a major revision (compared to v1): 1) we considered more general weights, i. e. $\theta_m= (\log m)^j m^\alpha$, 2) title replaced, 3) improvements of the presentation, 4) correction of typos and minor mathematical error

Large cycles and a functional central limit theorem for generalized weighted random permutations

The objects of our interest are the so-called $A$-permutations, which are permutations whose cycle length lie in a fixed set $A$. They have been extensively studied with respect to the uniform or the Ewens measure. In this paper, we extend some classical results to a more general weighted probability measure which is a natural extension of the Ewens measure and which in particular allows to consider sets $A_n$ depending on the degree $n$ of the permutation. By means of complex analysis arguments and under reasonable conditions on generating functions we study the asymptotic behaviour of classical statistics. More precisely, we generalize results concerning large cycles of random permutations by Vershik, Shmidt and Kingman, namely the weak convergence of the size ordered cycle length to a Poisson-Dirichlet distribution. Furthermore, we apply our tools to the cycle counts and obtain a Brownian motion central limit theorem which extends results by DeLaurentis, Pittel and Hansen.Comment: 24 pages, 3 Figure

Random permutation matrices under the generalized Ewens measure

We consider a generalization of the Ewens measure for the symmetric group, calculating moments of the characteristic polynomial and similar multiplicative statistics. In addition, we study the asymptotic behavior of linear statistics (such as the trace of a permutation matrix or of a wreath product) under this new measure.Comment: Published in at http://dx.doi.org/10.1214/12-AAP862 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fluctuations near the limit shape of random permutations under a conservative measure

In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set {1, ... n} under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process