Conclusion

This project has provided some insight into what happens to the number of eigenvalues of permutation matrices lying in a shrinking interval, but there are still several questions about $X_{n,a}$ that remain unanswered. One open question is the variance of $X_{n,a}$. As one might expect, calculating the variance turns out to be more complicated than calculating the mean. However, some preliminary calculations have shown that for $a = 0$, $Var[X_{n,a}]$ remains bounded as $n\rightarrow\infty$, and we suspect that it will be bounded for other values of $a$ as well. It would be interesting to try to derive a formula for the limit of the variance, and more generally, to be able to describe the distribution of $X_{n,a}$ in greater detail.