The Annals of Probability

Eigenvalue distributions of random permutation matrices

Kelly Wieand

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Let $M$ be a randomly chosen $n \times n$ permutation matrix. For a fixed arc of the unit circle, let $X$ be the number of eigenvalues of $M$ which lie in the specified arc. We calculate the large $n$ asymptotics for the mean and variance of $X$, and show that $(X -E[X])/( \Var (X))^ 1/2$ is asymptotically normally distributed. In addition, we show that for several fixed arcs $I_1,\ldots,I_m$, the corresponding random variables are jointly normal in the large $n$ limit.

Article information

Ann. Probab. Volume 28, Number 4 (2000), 1563-1587.

First available: 18 April 2002

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Zentralblatt MATH identifier

Primary: 15A52 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60F05: Central limit and other weak theorems 60C05

Permutations random matrices


Wieand, Kelly. Eigenvalue distributions of random permutation matrices. The Annals of Probability 28 (2000), no. 4, 1563--1587. doi:10.1214/aop/1019160498.

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