Permutation Matrices and the Moments of their Characteristic Polynomials

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 $\E{\prod_{k=1}^p Z_n^{s_k}(x_k)}$ 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 $Z_n^s(x)\stackrel{d}{\to }{Z}_{\infty }^s(x)$ 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 $n\rw\infty$.

arXiv comment24 pages, 1 Figure

References in corpus (1)

• On the averages of characteristic polynomials from classical groups

Cited by in corpus (1)

• On averages of randomized class functions on the symmetric groups and their asymptotics