  Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge
theory of the symmetric group S(n) defined on a cell discretization of the
surface. We study the theory in the large-n limit, and we find a rich phase
diagram with first and second order transition lines. The various phases are
characterized by different connectivity properties of the covering surface. We
point out some interesting connections with the theory of random walks on group
manifolds and with random graph theory.
