  We study the two-dimensional gauge theory of the symmetric group S_n
describing the statistics of branched n-coverings of Riemann surfaces. We
consider the theory defined on the disk and on the sphere in the large-n limit.
A non trivial phase structure emerges, with various phases corresponding to
different connectivity properties of the covering surface. We show that any
gauge theory on a two-dimensional surface of genus zero is equivalent to a
random walk on the gauge group manifold: in the case of S_n, one of the phase
transitions we find can be interpreted as a cutoff phenomenon in the
corresponding random walk. A connection with the theory of phase transitions in
random graphs is also pointed out. Finally we discuss how our results may be
related to the known phase transitions in Yang-Mills theory. We discover that a
cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere,
in a large N limit where the coupling constant is scaled with N with an extra
log N compared to the standard 't Hooft scaling.
