### 4.13 Exploring geometric properties

Since the effective geometry in both phases is rather singular, different ways of measuring
length may lead to inequivalent definitions of “dimension”. A common (local) notion is derived
from the volume of a geodesic ball with radius . Usually the radius is measured in terms of
the geodesic distance (the minimal number of links), or dual geodesic distance (the
minimal number of links of the dual graph). Alternatively, one may consider the number of
4-simplices in spherical shells of thickness 1 at distance , and define a fractal dimension by
[90, 14].
To extract a global length scale, one may use the averages , , or consider the average “radius
of the universe” [11] to obtain a cosmological Hausdorff dimension , or the “average intrinsic
linear extent” (see, for example, [75, 76]). Away from the phase transition, the
fractal dimensions associated with these geometric construction are more or less equivalent and give
in the crumpled phase and in the elongated phase. It is difficult to measure the
dimension close to the transition point.

It will not be straightforward to interpret the behaviour of observables (defined in analogy
with the continuum theory), since in most of the phase space the geometry of the simplicial
complex is far from approximating a metric 4-manifold. In search of a semiclassical interpretation
for geometric observables, an alternative notion of local curvature for a simplicial manifold
was suggested by de Bakker and Smit [90], based on a continuum expansion of the volume
of a geodesic ball. Assuming furthermore that independent of , -volumes of balls with
radius behave like regions on , they extracted scaling relations for various
geometric quantities for an intermediate range for . This line of thought was pursued further
in [185].

Close to the phase transition, one may investigate the behaviour of test particles (ignoring
back-reactions on the geometry). Comparing the mass extracted from the one-particle propagator with the
energy of the combined system obtained from the two-particle propagator [88, 92, 93], one does indeed find
evidence for gravitational binding.