Electronic Journal of Probability

Eigenvalues of Random Wreath Products

Steven Evans

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Consider a uniformly chosen element $X_n$ of the $n$-fold wreath product $\Gamma_n = G \wr G \wr \cdots \wr G$, where $G$ is a finite permutation group acting transitively on some set of size $s$. The eigenvalues of $X_n$ in the natural $s^n$-dimensional permutation representation (the composition representation) are investigated by considering the random measure $\Xi_n$ on the unit circle that assigns mass $1$ to each eigenvalue.  It is shown that if $f$ is a trigonometric polynomial, then  $\lim_{n \rightarrow \infty} P\{\int f d\Xi_n \ne s^n \int f d\lambda\}=0$, where $\lambda$ is normalised Lebesgue measure on the unit circle. In particular, $s^{-n} \Xi_n$ converges weakly in probability to $\lambda$ as $n \rightarrow \infty$.  For a large class of test functions $f$ with non-terminating Fourier expansions, it is shown that there exists a constant $c$ and a non-zero random variable $W$ (both depending on $f$) such that $c^{-n} \int f d\Xi_n$ converges in distribution as $n \rightarrow \infty$ to $W$.  These results have applications to Sylow $p$-groups of symmetric groups and autmorphism groups of regular rooted trees.

Article information

Electron. J. Probab., Volume 7 (2002), paper no. 9, 15 pp.

Accepted: 2 April 2002
First available in Project Euclid: 16 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52
Secondary: 05C05: Trees 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

random permutation random matrix Haar measure regular tree Sylow branching process multiplicative function

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Evans, Steven. Eigenvalues of Random Wreath Products. Electron. J. Probab. 7 (2002), paper no. 9, 15 pp. doi:10.1214/EJP.v7-108. https://projecteuclid.org/euclid.ejp/1463434882

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  • M. Abert and P. Diaconis. One-two-tree(s) and Sylow subgroups of S_n. Technical report, Department of Mathematics, Stanford University, 2002.
  • M. Abert and B. Virag. Groups acting on regular trees: probability and Hausdorff dimension. Technical report, Department of Mathematics, M.I.T., 2002.
  • R.A. Bailey, Cheryl E. Praeger, C.A. Rowley, and T.P. Speed. Generalized wreath products of permutation groups. Proc. London Math. Soc. (3), 47:69-82, 1983.
  • Hyman Bass, Maria Victoria Otero-Espinar, Daniel Rockmore, and Charles Tresser. Cyclic renormalization and automorphism groups of rooted trees. Springer-Verlag, Berlin 1996.
  • Persi Diaconis and Steven N. Evans. Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc., 353(7):2615-2633, 2001.
  • P. Diaconis and M. Shahshahani. On the eigenvalues of random matrices. J. Appl. Probab., 31A:49-62, 1994.
  • Knut Dale and Ivar Skau. The (generalized) secretary's packet problem and the Bell numbers. Discrete Math., 137(1-3):357-360, 1995.
  • A. Dyubina. Characteristics of random walks on the wreath products of groups. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 256 (Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 3): 31-37, 264. 1999.
  • A. G. Dyubina. An example of the rate of departure to infinity for a random walk on a group. Uspekhi Mat. Nauk, 54(5(329)):159-160, 1999.
  • James Allen Fill and Clyde H. Schoolfield, Jr. Mixing times for Markov chains on wreath products and related homogeneous spaces. Electron. J. Probab., 6: no. 11, 22 pp., 2001.
  • C. P. Hughes, J. P. Keating, and Neil O'Connell. On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys., 220(2):429-451, 2001.
  • B. M. Hambly, P. Keevash, N. O'Connell, and D. Stark. The characteristic polynomial of a random permutation matrix. Stochastic Process. Appl., 90(2):335-346, 2000.
  • Gordon James and Adalbert Kerber. The representation theory of the symmetric group. Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn, With an introduction by Gilbert de B. Robinson.
  • K. Johansson. On random matrices from the compact classical groups. Ann. of Math. (2), 145:519-545, 1997.
  • Adalbert Kerber. Representations of permutation groups. I. Springer-Verlag, Berlin, 1971. Lecture Notes in Mathematics, Vol. 240.
  • Adalbert Kerber. Representations of permutation groups. II. Springer-Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 495.
  • J. P. Keating and N. C. Snaith. Random matrix theory and L-functions at s=1/2. Comm. Math. Phys., 214(1):91-110, 2000.
  • J. P. Keating and N. C. Snaith. Random matrix theory and zeta(1/2+it). Comm. Math. Phys., 214(1):57-89, 2000.
  • V. A. Kaimanovich and A. M. Vershik. Random walks on discrete groups: boundary and entropy. Ann. Probab., 11(3):457-490, 1983.
  • H.C. Longuet-Higgins. The symmetry groups of non-rigid molecules. Molecular Physics, 6:445-460, 1963.
  • Russell Lyons, Robin Pemantle, and Yuval Peres. Random walks on the lamplighter group. Ann. Probab., 24(4):1993-2006, 1996.
  • Madan Lal Mehta. Random matrices. Academic Press Inc., Boston, MA, second edition, 1991.
  • P. P. Palfy and M. Szalay. The distribution of the character degrees of the symmetric p-groups. Acta Math. Hungar., 41(1-2):137-150, 1983.
  • P. P. Palfy and M. Szalay. On a problem of P. Turan concerning Sylow subgroups. In Studies in pure mathematics, pages 531-542. Birkhauser, Basel, 1983.
  • P. P. Palfy and M. Szalay. Further probabilistic results on the symmetric p-groups. Acta Math. Hungar., 53(1-2):173-195, 1989.
  • Christophe Pittet and Laurent Saloff-Coste. Amenable groups, isoperimetric profiles and random walks. In Geometric group theory down under (Canberra, 1996), pages 293-316. de Gruyter, Berlin, 1999.
  • E.M. Rains. High powers of random elements of compact Lie groups. Probab. Theory Related Fields, 107:219-241, 1997.
  • K.L. Wieand. Eigenvalue distributions of random matrices in the permutation group and compact Lie groups. PhD thesis, Harvard University, 1998.
  • Kelly Wieand. Eigenvalue distributions of random permutation matrices. Ann. Probab., 28(4):1563-1587, 2000.