## 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.

### MSC:

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

### Keywords:

permutations; random matrices
Full Text:

### References:

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