Permutation Matrices and the Moments of Their Characteristics Polynomials

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 [ ∏p k=1 Z sk n (xk) ] for sk ∈ N. We show with this generating function that limn→∞ E [ ∏p k=1 Z sk n (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 Z ∞ (x) such that Z n(x) d −→ Z ∞ (x) and E [ ∏p k=1 Z sk n (xk) ] → E [ ∏p k=1 Z sk ∞ (xk) ] for maxk |xk| < 1 and n → ∞.