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## The Number of Eigenvalues at

When random points are chosen uniformly on the unit circle, the probability of picking any particular point is 0. The eigenvalues of permutation matrices, however, occur only at certain values of , so the probability of choosing one of these points is positive, while the probability for any other point is 0.

In order to gain some insight into this problem, define a random variable to be the number of eigenvalues of equal to . The variable is already well understood; see, for example, [1], [2]. Presented here is a brief explanation of what happens to the mean of as .

First consider the case when . Every cycle in a permutation produces the eigenvalue 1, so the number of eigenvalues at will equal the total number of cycles in . Recall that the number of cycles in is the sum of all the values of Ck in the cycle structure. Thus,

 (14)

and the mean of Zn,0 is

 (15)

For large n, this sum can be approximated by , resulting in

 (16)

Similar reasoning can be used to see that in general, if with p and qrelatively prime, then

 (17)

If is an irrational multiple of , then no eigenvalues can occur there, so for all n in this case. This behavior is quite different from the uniform case.

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2000-09-25