TY - JOUR

T1 - Permutation matrices and the moments of their characteristics polynomials

AU - Zeindler, Dirk

N1 - This work is licensed under a Creative Commons Attribution 3.0 License.

PY - 2010

Y1 - 2010

N2 - 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[∏pk=1Zskn(xk)] for sk∈N. We show with this generating function that limn→∞E[∏pk=1Zskn(xk)] exists 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)→dZs∞(x) and E[∏pk=1Zskn(xk)]→E[∏pk=1Zsk∞(xk)] for maxk|xk|<1 and n→∞.

AB - 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[∏pk=1Zskn(xk)] for sk∈N. We show with this generating function that limn→∞E[∏pk=1Zskn(xk)] exists 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)→dZs∞(x) and E[∏pk=1Zskn(xk)]→E[∏pk=1Zsk∞(xk)] for maxk|xk|<1 and n→∞.

U2 - 10.1214/EJP.v15-781

DO - 10.1214/EJP.v15-781

M3 - Journal article

VL - 15

SP - 1092

EP - 1118

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

ER -