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2010 Permutation Matrices and the Moments of their Characteristics Polynomials
Dirk Zeindler
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Electron. J. Probab. 15: 1092-1118 (2010). DOI: 10.1214/EJP.v15-781

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 $n\to\infty$.

Citation

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Dirk Zeindler. "Permutation Matrices and the Moments of their Characteristics Polynomials." Electron. J. Probab. 15 1092 - 1118, 2010. https://doi.org/10.1214/EJP.v15-781

Information

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

zbMATH: 1225.15038
MathSciNet: MR2659758
Digital Object Identifier: 10.1214/EJP.v15-781

Subjects:
Primary: 15B52

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Vol.15 • 2010
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