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Eigenvalues of Random Wreath Products
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Steven N. Evans, University of California at Berkeley
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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 $lim_{n rightarrow infty} P{int f dXi_n ne s^n int f
dlambda}=0$, 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 $c^{-n} int f dXi_n$ 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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Full text: PDF
Pages: 1-15
Published on: July 31, 2001
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Electronic Journal of Probability. ISSN: 1083-6489 |
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