Electronic Journal of Probability

Eigenvalues of Random Wreath Products

Steven Evans

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1463434882

Digital Object Identifier
doi:10.1214/EJP.v7-108

Mathematical Reviews number (MathSciNet)
MR1902842

Zentralblatt MATH identifier
1013.15006

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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