Since the largest cycle that can occur in
is an *n*-cycle, the first position on
the unit circle where an eigenvalue can occur is at
.
If *l*<1, then
*X*_{n,0} = 0 because the interval ends before reaching the first possible eigenvalue. This
can also be seen from (18) by noting that
if *l*<1.

Now if *l*=1, then the interval ends exactly where the first eigenvalue can occur, so we
have

The probability that has an

and

Both the mean and the variance approach 0 as .

**Remark.** A similar analysis shows that an analogous `gap' occurs around each
rational point *a*. In particular, if *a* = *p*/*q* in lowest terms, then no eigenvalues can fall
in the interval *I*_{n} if *l*<1/*q*. For irrational values of *a*, this sort of gap does not
occur. No matter how small *l* is, there always will be some values of *n* that produce
eigenvalues in the interval *I*_{n}.

These results illuminate some of the differences between the distribution of eigenvalues and
that of independent points on the circle. Equations (19) and (20) describe a particularly simple situation, and in this case, the results for random
permutations are strikingly different from the results for random independent points. When
*l*>1, the simple argument used here no longer works, and it may not be possible to find
explicit formulas for
*E*[*X*_{n,a}] or
*Var*[*X*_{n,a}] in terms of *n*. In that case, when *n* is small,
it is easy to calculate the value of *X*_{n,a}, and of
*E*[*X*_{n,a}] or other quantities describing the
distribution of eigenvalues. As *n* increases, however, exact results require more and more
computation, and it is more useful to try to find general trends that will provide a picture
of what is happening. The following sections use a different approach to find the large *n*limit of
*E*[*X*_{n,a}], and the goal will be to see whether this limit might resemble the
independent points case more closely when the constant *l* is larger.