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Eigenvalues of Random Wreath Products

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Eigenvalues of Random Wreath Products
2.	Creator	Author's name, affiliation, country	Steven N. Evans; University of California at Berkeley
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	  random permutation, random matrix, Haar measure, regular tree, Sylow, branching process, multiplicative function
3.	Subject	Subject classification	15A52, 05C05, 60B15, 60J80
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2002-04-02
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/108
10.	Identifier	Digital Object Identifier	10.1214/EJP.v7-108
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 7
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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