  We use the large deviation approach to sum rules pioneered by Gamboa, Nagel
and Rouault to prove higher order sum rules for orthogonal polynomials on the
unit circle. In particular, we prove one half of a conjectured sum rule of
Lukic in the case of two singular points, one simple and one double. This is
important because it is known that the conjecture of Simon fails in exactly
this case, so this paper provides support for the idea that Lukic's replacement
for Simon's conjecture might be true.
