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.
Zeindler, D