An Investigation of the Number of Eigenvalues

Lying in a Shrinking Interval

**Nathaniel Blair-Stahn**

When an
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.

- Introduction
- Background About Permutations and Probability
- Permutation Matrices and
*X*_{n,a} - Preliminary Observations
- A Technical Lemma
- Calculation of the Mean
- Conclusion
- Bibliography
- About this document ...