  We give a rigorous proof of the fact that a phase transition discovered by
Douglas and Kazakov in 1993 in the context of two-dimensional gauge theories
occurs. This phase transition can be formulated in terms of the Brownian bridge
on the unitary group U(N) when N tends to infinity. We explain how it can be
understood by considering the asymptotic behaviour of the eigenvalues of the
unitary Brownian bridge, and how it can be technically approached by means of
Fourier analysis on the unitary group. Moreover, we advertise some more or less
classical methods for solving certain minimisation problems which play a
fundamental role in the study of the phase transition.
