Permutation Matrices and the Moments of their Characteristic Polynomials

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 $\text{E}{\prod }_{k=1}^p{Z}_n^{s_k}(x_k)$ for $s_k\in \text{Nr}$. We show with this generating function that $\lim_{n\rw\infty} \E{\prod_{k=1}^p Z_n^{s_k}(x_k)}$existsfor$\max_k|x_k|<1$andcalculatethegrowthratefor$p=2, |x_1|=|x_2|=1$,$x_1=\overline{x_2}$and$n\rw\infty$. We also look at the case $s_k\in \text{C}$. We use the Feller coupling to show that for each $\mathrm{\mid }x\mathrm{\mid }<1$ and $s\in \text{C}$ there exists a random variable $Z_{\mathrm{\infty }}^s(x)$ such that $Z_n^s(x)\stackrel{d}{\to }{Z}_{\mathrm{\infty }}^s(x)$ 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$n\rw\infty$.