Eigenvalue distributions of random permutation matrices. (English) Zbl 1044.15017

Summary: 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 \(I_1,\dots,I_m\), the corresponding random variables are jointly normal in the large \(n\) limit.


15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
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