Eigenvalues of random wreath products. (English) Zbl 1013.15006

Evans, Steven N.

Electron. J. Probab. 7, Paper No. 9, 15 p. (2002).

The author considers the limiting distribution of eigenvalues of random elements of the \(n\)-fold wreath product of a finite transitive permutation group. It is proved that there is weak convergence in probability to the standard Lebesgue measure on the unit circle. This is a corollary of a stronger result that the limiting probability that expectation of trigonometic polynomial is not its expectation under Lebesgue measure is zero. The result is extended to a class of infinite Fourier series.

Reviewer: Ki Hang Kim (Montgomery)

MSC:

15B52	Random matrices (algebraic aspects)
42A20	Convergence and absolute convergence of Fourier and trigonometric series
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
15A18	Eigenvalues, singular values, and eigenvectors
05C05	Trees
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization

Keywords:

random permutation; random matrix; Haar measure; regular tree; Sylow; branching process; multiplicative function; eigenvalues; wreath product; finite transitive permutation group; weak convergence; Lebesgue measure; infinite Fourier series