  We discuss a general mechanism that drives the phase transition in the
canonical ensemble in models of random geometries. As an example we consider a
solvable model of branched polymers where the transition leading from tree- to
bush-like polymers relies on the occurrence of vertices with a large number of
branches. The source of this transition is a combination of the constraint on
the total number of branches in the canonical ensemble and a nonlinear
one-vertex action. We argue that exactly the same mechanism, which we call
constrained mean-field, plays the crucial role in the phase transition in 4d
simplicial gravity and, when applied to the effective one-vertex action,
explains the occurrence of both the mother universe and singular vertices at
the transition point when the system enters the crumpled phase.
