216 research outputs found
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 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 . 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
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