Eigenvalue distributions of random permutation matrices

Abstract

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 $(X -E[X])/( \text{Var} (X))^ {1/2}$ 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.