Permutation matrices and the moments of their characteristic polynomial

Abstract

In this paper, we are interested in the moments of the characteristic polynomial Zn(x) of the n× n permutation matrices with respect to the uniform measure. We use a combinatorial argument to write down the generating function of 𝔼[Πpk=1 Zskn(xk)] for Sk ∈ ℕ. We show with this generating function that limn → ∞ 𝔼[Πpk=1 Zskn(xk)] exists for maxk|xk| < 1 and calculate the growth rate for p = 2, |x1| = |x2| = 1, x1 = x2¯ and n → ∞. We also look at the case sk ∈ ℂ. We use the Feller coupling to show that for each |x| < 1 and s ∈ ℂ there exists a random variable Zs∞(x) such that Zsn(x) →d Zs∞(x) 𝔼[Πpk=1 Zskn(xk)] → 𝔼[Πpk=1 Zsk∞(xk)] for maxk|xk| < 1 and n → ∞. © 2010 Applied Probability Trust.