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Permutation Matrices and the Moments of their Characteristics Polynomials

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1.	Title	Title of document	Permutation Matrices and the Moments of their Characteristics Polynomials
2.	Creator	Author's name, affiliation, country	Dirk Zeindler; University Zürich; Switzerland
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	random permutation matrices, symmetric group, characteristic polynomials, Feller coupling, asymptotic behavior of moments, generating functions
3.	Subject	Subject classification	15B52
4.	Description	Abstract	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 $E[\prod_{k=1}^pZ_n^{s_k}(x_k)]$ for $s_k\in\mathbb{N}$. We show with this generating function that $\lim_{n\to\infty}E[\prod_{k=1}^pZ_n^{s_k}(x_k)]$ exists exists for $\max_k|x_k|<1$ and calculate the growth rate for $p=2$, $|x_1|=|x_2|=1$, $x_1=x_2$ and $n\to\infty$. We also look at the case $s_k\in\mathbb{C}$. We use the Feller coupling to show that for each $|x|<1$ and $s\in\mathbb{C}$ there exists a random variable $Z_\infty^s(x)$ such that $Z_n^s(x)\overset{d}{\to}Z_\infty^s(x)$ and $E[\prod_{k=1}^pZ_n^{s_k}(x_k)]\to E[\prod_{k=1}^pZ_\infty^{s_k}(x_k)]$ for $\max_k|x_k|<1$ and $n\to\infty$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2010-07-07
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/781
10.	Identifier	Digital Object Identifier	10.1214/EJP.v15-781
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 15
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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