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Random Permutation Matrices
An Investigation of the Number of Eigenvalues
Lying in a Shrinking Interval

Nathaniel Blair-Stahn

Abstract:

When an $n \times n$ permutation matrix is chosen at random, each of its n eigenvalues will lie somewhere on the unit circle. We investigate the average number of these that fall in an interval that shrinks as the size of the matrix increases, and compare the results against the case where n points are chosen independently.