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 Electronic Journal of Probability > Vol. 15(2010) > Paper 34 open journal systems 


Permutation matrices and the moments of their characteristics polynomials

Dirk Zeindler, University Zürich


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


Full text: PDF

Pages: 1092-1118

Published on: July 7, 2010
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Electronic Journal of Probability. ISSN: 1083-6489