| Título: | Permutation Matrices and the Moments of their Characteristics Polynomials |
| Autores: | Zeindler, Dirk; University Zürich |
| Fecha: | 2010-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: |
Peer-reviewed Article |
| Tema: |
random permutation matrices, symmetric group, characteristic polynomials, Feller coupling, asymptotic behavior of moments, generating functions 15B52 |
| Descripción: | 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 $n\to\infty$. |
| Idioma: | No aplica |
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