Permutation matrices and the moments of their characteristics polynomials. (English) Zbl 1225.15038

Summary: 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 \(\mathbb E[\prod_{k=1}^{p} Z_{n}^{s_{k}}(x_{k})]\) for \(s_{k}\in\mathbb N\). We show with this generating function that \(\lim_{n\to\infty}\mathbb E[\prod_{k=1}^{p} Z_n^{s_k}(x_k)]\) 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 \(n\to \infty\).
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 \(Z_n^s(x)\overset {d} \rightarrow Z_\infty^s(x)\) and \(\mathbb E[\prod_{k=1}^p Z_n^{s_k}(x_k)]\to\mathbb E[\prod_{k=1}^p Z_\infty^{s_k}(x_k)]\) for \(\max_k|x_k|<1\) and \(n\to\infty\).


15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
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