Appearance of mother universe and singular vertices in random geometries

We discuss a general mechanism that drives the phase transition in a wide class of models, including those of random geometries. As an example we consider a solvable model of branched polymers which undergoes a transition from tree-like to bush-like polymers. The source of this transition is a combination of the constraint on the average number of branches per vertex and a non-linear 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. (C) 1997 Elsevier Science B.V.