### 4.15 Summary

In the dynamical triangulations approach, one studies the properties of a statistical ensemble of
simplicial four-geometries à la Regge of fixed edge lengths. By summing over such discrete configurations
according to (21), one has implicitly assumed that this leads to a uniform sampling of the space of smooth
Riemannian manifolds. There is no obvious weak-field limit, but this is no obstacle in principle to the
path-integral construction. Numerical simulations indicate the existence of a well-defined phase for
sufficiently small (inverse Newton’s constant) and a sufficiently large cosmological constant. For small
, one finds a “crumpled” phase, with small average curvature and a large Hausdorff dimension, and for
large an elongated, effectively two-dimensional polymer phase. At present, the consensus seems to be
that the corresponding phase transition is of first order, with a finite average curvature at the transition
point.
Almost all simulations have been done on simplicial manifolds with -topology. Neither the inclusion
of factors of in the measure nor the addition of higher-order curvature terms to the action seem
to have a substantial influence on the phase structure. Also matter coupling to spinorial and
scalar fields does not seem to lead to a change of universality class, although the inclusion of
several gauge fields may have a more drastic effect. The study of singular structures (vertices of
high coordination number) has led to a qualitative understanding of the phase structure of the
model.