# Permutation matrices and the moments of their characteristic polynomial

Zeindler, D (2010). Permutation matrices and the moments of their characteristic polynomial. Electronic Journal of Probability, 15(34):1092-1118.

## Abstract

In this paper, we are interested in the moments of the characteristic polynomial Z(n)(x) of the n x n permutation matrices with respect to the uniform measure. We use a combinatorial argument to write down the generating function of E [Pi(p)(k=1) Z(n)(sk)(x(k))] for s(k) is an element of N. We show with this generating function that lim(n ->infinity) E [Pi(p)(k=1) Z(n)(sk)(x(k))] exists for max(k) vertical bar x(k)vertical bar < 1 and calculate the growth rate for p = 2, vertical bar x(1)vertical bar = vertical bar x(2)vertical bar = 1, x(1) = x(2) and n -> infinity.

We also look at the case s(k) is an element of C. We use the Feller coupling to show that for each vertical bar x vertical bar < 1 and s is an element of C there exists a random variable Z(infinity)(s)(x) such that Z(n)(s)(x)(d) -> Z(infinity)(s)(x) and E[Pi(p)(k=1) Z(n)(sk)(x(k))] -> E [Pi(p)(k=1) Z(infinity)(sk)(x(k))] for max(k) vertical bar xk vertical bar < 1 and n -> infinity.

## Abstract

In this paper, we are interested in the moments of the characteristic polynomial Z(n)(x) of the n x n permutation matrices with respect to the uniform measure. We use a combinatorial argument to write down the generating function of E [Pi(p)(k=1) Z(n)(sk)(x(k))] for s(k) is an element of N. We show with this generating function that lim(n ->infinity) E [Pi(p)(k=1) Z(n)(sk)(x(k))] exists for max(k) vertical bar x(k)vertical bar < 1 and calculate the growth rate for p = 2, vertical bar x(1)vertical bar = vertical bar x(2)vertical bar = 1, x(1) = x(2) and n -> infinity.

We also look at the case s(k) is an element of C. We use the Feller coupling to show that for each vertical bar x vertical bar < 1 and s is an element of C there exists a random variable Z(infinity)(s)(x) such that Z(n)(s)(x)(d) -> Z(infinity)(s)(x) and E[Pi(p)(k=1) Z(n)(sk)(x(k))] -> E [Pi(p)(k=1) Z(infinity)(sk)(x(k))] for max(k) vertical bar xk vertical bar < 1 and n -> infinity.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Statistics and Probability Social Sciences & Humanities > Statistics, Probability and Uncertainty 15B52, random permutation matrices, symmetric group, characteristic polynomials, Feller coupling, asymptotic behavior of moments, generating functions English July 2010 16 Aug 2010 14:40 29 Jul 2020 22:51 Institute of Mathematical Statistics 1083-6489 Hybrid https://doi.org/10.1214/EJP.v15-781 http://www.math.washington.edu/~ejpecp/viewarticle.php?id=2097&layout=abstract http://arxiv.org/abs/0910.5069