Addendum: Level Spacings for Integrable Quantum Maps in Genus Zero

In this addendum we strengthen the results of math-ph/0002010 in the case of polynomial phases. We prove that Cesaro means of the pair correlation functions of certain integrable quantum maps on the 2-sphere at level N tend almost always to the Poisson (uniform limit). The quantum maps are exponentials of Hamiltonians which have the form a p(I) + b I, where I is the action, where p is a polynomial and where a,b are two real numbers. We prove that for any such family and for almost all a,b, the pair correlation tends to Poisson on average in N. The results involve Weyl estimates on exponential sums and new metric results on continued fractions. They were motivated by a comparison of the results of math-ph/0002010 with some independent results on pair correlation of fractional parts of polynomials by Rudnick-Sarnak.Comment: Addendum to math-ph/000201

Macdonald's identities and the large N limit of YM2YM_2 on the cylinder

We give a rigorous calculation of the large N limit of the partition function of SU(N) gauge theory on a 2D cylinder in the case where one boundary holomony is a so-called special element of type $\rho$. By MacDonald's identity, the partition function factors in this case as a product over positive roots and it is straightforward to calculate the large N asymptotics of the free energy. We obtain the unexpected result that the free energy in these cases is asymptotic to N times a functional of the limit densities of eigenvalues of the boundary holonomies. This appears to contradict the predictions of Gross-Matysin and Kazakov-Wynter that the free energy should have a limit governed by the complex Burgers equation