## Electronic Journal of Probability

### Permutation Matrices and the Moments of their Characteristics Polynomials

Dirk Zeindler

#### 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 $n\to\infty$.

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 34, 1092-1118.

Dates
Accepted: 7 July 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819819

Digital Object Identifier
doi:10.1214/EJP.v15-781

Mathematical Reviews number (MathSciNet)
MR2659758

Zentralblatt MATH identifier
1225.15038

Subjects
Primary: 15B52: Random matrices

Rights

#### Citation

Zeindler, Dirk. Permutation Matrices and the Moments of their Characteristics Polynomials. Electron. J. Probab. 15 (2010), paper no. 34, 1092--1118. doi:10.1214/EJP.v15-781. https://projecteuclid.org/euclid.ejp/1464819819

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