arXiv 0910.50692009-10math.PRmath.NT

置换矩阵及其特征多项式的矩

Permutation Matrices and the Moments of their Characteristic Polynomials

Dirk Zeindler

备注：: 24 pages, 1 Figure
发表信息：: Electronic Journal of Probability,volume 15, 2010, pages 1092-1118, http://www.math.washington.edu/~ejpecp/EjpVol15/paper34.abs.html

摘要：

本文研究了$n\times n$置换矩阵的特征多项式$Z_n(x)$关于一致测度的矩。我们使用一个组合参数来写出$s_k\in\Nr$的$\E{\prod_{k=1}^p Z_n^{s_k}(x_k)}$的生成函数。我们证明了这个生成函数，$\lim_{n\rw\infty} \E{\prod_{k=1}^p Z_n^{s_k}(x_k)}$存在的$\max_k|x_k|<1$和计算的增长率为$p=2, |x_1|=|x_2|=1$，$x_1=\overline{x_2}$和$n\rw\infty$。我们也看看$s_k\in\C$的情况。我们使用Feller耦合表明，对于每个$|x|<1$和$s\in\C$，存在一个随机变量$Z_\infty^s(x)$，使得对于$\max_k|x_k|<1$和$n\rw\infty$，$Z_n^s(x)\xrightarrow{d}Z_\infty^s(x)$和$\E{\prod_{k=1}^p Z_n^{s_k}(x_k)}\rw \E{\prod_{k=1}^p Z_\infty^{s_k}(x_k)}$。

英文摘要：

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 $Z_n^s(x)\xrightarrow{d}Z_\infty^s(x)$ 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 $n\rw\infty$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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