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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 Zs(x) such that Zns(x) →d Zs(x) and E[∏k=1p Znsk(xk)]→E[∏k=1p Zsk(xk)] for maxk|xk| < 1 and n→∞


Full text: PDF

Pages: 1092-1118

Published on: July 7, 2010


Bibliography
  1. Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2003. xii+363 pp. ISBN: 3-03719-000-0 MR2032426 (2004m:60004)
  2. Bourgade, P.; Hughes, C. P.; Nikeghbali, A.; Yor, M. The characteristic polynomial of a random unitary matrix: a probabilistic approach. Duke Math. J. 145 (2008), no. 1, 45--69. MR2451289 (2009j:60011)
  3. Bump, Daniel. Lie groups. Graduate Texts in Mathematics, 225. Springer-Verlag, New York, 2004. xii+451 pp. ISBN: 0-387-21154-3 MR2062813 (2005f:22001)
  4. Bump, Daniel; Gamburd, Alex. On the averages of characteristic polynomials from classical groups. Comm. Math. Phys. 265 (2006), no. 1, 227--274. MR2217304 (2008f:60009)
  5. Flajolet, Philippe; Sedgewick, Robert. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp. ISBN: 978-0-521-89806-5 MR2483235 (Review)
  6. Freitag, Eberhard; Busam, Rolf. Complex analysis. Translated from the 2005 German edition by Dan Fulea. Universitext. Springer-Verlag, Berlin, 2005. x+547 pp. ISBN: 978-3-540-25724-0; 3-540-25724-1 MR2172762 (2006g:30002)
  7. Fritzsche, Klaus; Grauert, Hans. From holomorphic functions to complex manifolds. Graduate Texts in Mathematics, 213. Springer-Verlag, New York, 2002. xvi+392 pp. ISBN: 0-387-95395-7 MR1893803 (2003g:32001)
  8. Gut, Allan. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, 2005. xxiv+603 pp. ISBN: 0-387-22833-0 MR2125120 (2006a:60001)
  9. Hambly, B. M.; Keevash, P.; O'Connell, N.; Stark, D. The characteristic polynomial of a random permutation matrix. Stochastic Process. Appl. 90 (2000), no. 2, 335--346. MR1794543 (2002c:15041)
  10. Harro Heuser. Lehrbuch der Analysis Teil 1. B. G. Teubner, 10 edition, 1993. Math. Review number not available.
  11. Keating, J. P.; Snaith, N. C. Random matrix theory and $zeta(1/2+it)$. Comm. Math. Phys. 214 (2000), no. 1, 57--89. MR1794265 (2002c:11107)
  12. Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN: 0-19-853489-2 MR1354144 (96h:05207)
















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Electronic Journal of Probability. ISSN: 1083-6489