  In this paper we study the critical behaviour of the fully-connected
p-colours Potts model at the dynamical transition. In the framework of Mode
Coupling Theory (MCT), the time autocorrelation function displays a two step
relaxation, with two exponents governing the approach to the plateau and the
exit from it. Exploiting a relation between statics and equilibrium dynamics
which has been recently introduced, we are able to compute the critical slowing
down exponents at the dynamical transition with arbitrary precision and for any
value of the number of colours p. When available, we compare our exact results
with numerical simulations. In addition, we present a detailed study of the
dynamical transition in the large p limit, showing that the system is not
equivalent to a random energy model.
