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Preliminary Observations

When a permutation is picked with uniform probability from Sn and the eigenvalues of $M_\sigma$are plotted, the result is that n points on the unit circle have been chosen ``at random", in the sense that the outcome of this experiment is not known beforehand. Obviously, though, not every every point on the circle is equally likely to be picked. In fact, only a finite set of points is possible, and the probability of picking a particular point depends on its location.

Plotting the eigenvalues of a random $n \times n$ permutation matrix can be compared with plotting n independent points chosen uniformly from the set of all points on the unit circle. The purpose of this section is to summarize what happens for independent uniform points, and then to make a few quick observations about the eigenvalue distribution of permutation matrices, providing a brief comparison of the two situations.