## 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 $$n\to\infty$$.

### MSC:

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