Electronic Journal of Probability

Permutation Matrices and the Moments of their Characteristics Polynomials

Dirk Zeindler

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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 $n\to\infty$.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 34, 1092-1118.

Accepted: 7 July 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15B52: Random matrices

random permutation matrices symmetric group characteristic polynomials Feller coupling asymptotic behavior of moments generating functions

This work is licensed under aCreative Commons Attribution 3.0 License.


Zeindler, Dirk. Permutation Matrices and the Moments of their Characteristics Polynomials. Electron. J. Probab. 15 (2010), paper no. 34, 1092--1118. doi:10.1214/EJP.v15-781. https://projecteuclid.org/euclid.ejp/1464819819

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  • Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2003. xii+363 pp. ISBN: 3-03719-000-0.
  • Bourgade, P.; Hughes, C. P.; Nikeghbali, A.; Yor, M. The characteristic polynomial of a random unitary matrix: a probabilistic approach. Duke Math. J. 145 (2008), no. 1, 45–69.
  • Bump, Daniel. Lie groups. Graduate Texts in Mathematics, 225. Springer-Verlag, New York, 2004. xii+451 pp. ISBN: 0-387-21154-3
  • Bump, Daniel; Gamburd, Alex. On the averages of characteristic polynomials from classical groups. Comm. Math. Phys. 265 (2006), no. 1, 227–274.
  • Flajolet, Philippe; Sedgewick, Robert. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp. ISBN: 978-0-521-89806-5 (Review)
  • Freitag, Eberhard; Busam, Rolf. Complex analysis. Translated from the 2005 German edition by Dan Fulea. Universitext. Springer-Verlag, Berlin, 2005. x+547 pp. ISBN: 978-3-540-25724-0; 3-540-25724-1
  • Fritzsche, Klaus; Grauert, Hans. From holomorphic functions to complex manifolds. Graduate Texts in Mathematics, 213. Springer-Verlag, New York, 2002. xvi+392 pp. ISBN: 0-387-95395-7
  • Gut, Allan. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, 2005. xxiv+603 pp. ISBN: 0-387-22833-0
  • Hambly, B. M.; Keevash, P.; O'Connell, N.; Stark, D. The characteristic polynomial of a random permutation matrix. Stochastic Process. Appl. 90 (2000), no. 2, 335–346.
  • Harro Heuser. Lehrbuch der Analysis Teil 1. B. G. Teubner, 10 edition, 1993.
  • Keating, J. P.; Snaith, N. C. Random matrix theory and $zeta(1/2+it)$. Comm. Math. Phys. 214 (2000), no. 1, 57–89.
  • Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN: 0-19-853489-2