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2002 Eigenvalues of Random Wreath Products
Steven Evans
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Electron. J. Probab. 7: 1-15 (2002). DOI: 10.1214/EJP.v7-108


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.


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Steven Evans. "Eigenvalues of Random Wreath Products." Electron. J. Probab. 7 1 - 15, 2002.


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

zbMATH: 1013.15006
MathSciNet: MR1902842
Digital Object Identifier: 10.1214/EJP.v7-108

Primary: 15A52
Secondary: 05C05 , 60B15 , 60J80

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


Vol.7 • 2002
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