# Appearence of Mother Universe and Singular Vertices in Random Geometries

## TL;DR

This paper explores a universal mechanism called constrained mean-field that explains phase transitions in random geometries, including the emergence of a mother universe and singular vertices in 4D simplicial gravity.

## Contribution

It introduces the constrained mean-field mechanism as a unifying explanation for phase transitions in models of random geometries, linking branched polymers and 4D gravity.

## Key findings

- Identifies the role of large-branch vertices in phase transitions.
- Demonstrates the mechanism in a solvable branched polymer model.
- Explains the emergence of singular vertices and mother universe in 4D gravity.

## Abstract

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.