In this paper, we are interested in the moments of the characteristic polynomial $Z_n(x)$ of the $n\times n$ permutation matrices with respect to the uniform measure. We use a combinatorial argument to write down the generating function of $E[\prod_{k=1}^pZ_n^{s_k}(x_k)]$ for $s_k\in\mathbb{N}$. We show with this generating function that $\lim_{n\to\infty}E[\prod_{k=1}^pZ_n^{s_k}(x_k)]$ exists exists for $\max_k|x_k|<1$ and calculate the growth rate for $p=2$, $|x_1|=|x_2|=1$, $x_1=x_2$ and $n\to\infty$. We also look at the case $s_k\in\mathbb{C}$. We use the Feller coupling to show that for each $|x|<1$ and $s\in\mathbb{C}$ there exists a random variable $Z_\infty^s(x)$ such that $Z_n^s(x)\overset{d}{\to}Z_\infty^s(x)$ and $E[\prod_{k=1}^pZ_n^{s_k}(x_k)]\to E[\prod_{k=1}^pZ_\infty^{s_k}(x_k)]$ for $\max_k|x_k|<1$ and $n\to\infty$.
Electron. J. Probab.
15:
1092-1118
(2010).
DOI: 10.1214/EJP.v15-781
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. 1040.60001Arratia, 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. 1040.60001
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. 1155.15025 10.1215/00127094-2008-046 euclid.dmj/1221656862Bourgade, 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. 1155.15025 10.1215/00127094-2008-046 euclid.dmj/1221656862
Bump, Daniel. Lie groups. Graduate Texts in Mathematics, 225. Springer-Verlag, New York, 2004. xii+451 pp. ISBN: 0-387-21154-3 MR2062813 1053.22001Bump, Daniel. Lie groups. Graduate Texts in Mathematics, 225. Springer-Verlag, New York, 2004. xii+451 pp. ISBN: 0-387-21154-3 MR2062813 1053.22001
Bump, Daniel; Gamburd, Alex. On the averages of characteristic polynomials from classical groups. Comm. Math. Phys. 265 (2006), no. 1, 227–274. 1107.60004 10.1007/s00220-006-1503-1Bump, Daniel; Gamburd, Alex. On the averages of characteristic polynomials from classical groups. Comm. Math. Phys. 265 (2006), no. 1, 227–274. 1107.60004 10.1007/s00220-006-1503-1
Flajolet, Philippe; Sedgewick, Robert. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp. ISBN: 978-0-521-89806-5 (Review) MR2483235 1165.05001Flajolet, Philippe; Sedgewick, Robert. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp. ISBN: 978-0-521-89806-5 (Review) MR2483235 1165.05001
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 1085.30001Freitag, 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 1085.30001
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 1005.32002Fritzsche, 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 1005.32002
Gut, Allan. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, 2005. xxiv+603 pp. ISBN: 0-387-22833-0 1076.60001Gut, Allan. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, 2005. xxiv+603 pp. ISBN: 0-387-22833-0 1076.60001
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. 1047.60013 10.1016/S0304-4149(00)00046-6Hambly, 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. 1047.60013 10.1016/S0304-4149(00)00046-6
Keating, J. P.; Snaith, N. C. Random matrix theory and $zeta(1/2+it)$. Comm. Math. Phys. 214 (2000), no. 1, 57–89. 1051.11048 10.1007/s002200000261Keating, J. P.; Snaith, N. C. Random matrix theory and $zeta(1/2+it)$. Comm. Math. Phys. 214 (2000), no. 1, 57–89. 1051.11048 10.1007/s002200000261
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: 0-19-853489-2Macdonald, 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