29 research outputs found
Permutation Matrices and the Moments of their Characteristic Polynomials
In this paper, we are interested in the moments of the characteristic
polynomial of the 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 and calculate the growth rate for
, and n\rw\infty. We also look at
the case s_k\in\C. We use the Feller coupling to show that for each
and s\in\C there exists a random variable such that
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 and n\rw\infty.Comment: 24 pages, 1 Figur
The order of large random permutations with cycle weights
The order of a permutation of objects is the
smallest integer such that the -th iterate of 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 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 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. , 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 -permutations, which are
permutations whose cycle length lie in a fixed set . 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 depending on the degree 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