  Combinatorial quantum gravity is governed by a discrete Einstein-Hilbert
action formulated on an ensemble of random graphs. There is strong evidence for
a second-order quantum phase transition separating a random phase at strong
coupling from an ordered, geometric phase at weak coupling. Here we derive the
picture of space-time that emerges in the geometric phase, given such a
continuous phase transition. In the geometric phase, ground-state graphs are
discretizations of Riemannian, negative-curvature Cartan-Hadamard manifolds. On
such manifolds, diffusion is ballistic. Asymptotically, diffusion time is
soldered with a manifold coordinate and, consequently, the probability
distribution is governed by the wave equation on the corresponding Lorentzian
manifold of positive curvature, de Sitter space-time. With this asymptotic
Lorentzian picture, the original negative curvature of the Riemannian manifold
turns into a positive cosmological constant. The Lorentzian picture, however,
is valid only asymptotically and cannot be extrapolated back in coordinate
time. Before a certain epoch, coordinate time looses its meaning and the
universe is a negative-curvature Riemannian "shuttlecock" with ballistic
diffusion, thereby avoiding a big bang singularity. The so-called "dimension at
infinity" of negative curvature manifolds, i.e. the large-scale spectral
dimension seen by diffusion processes with no spectral gap, those that can
probe the geometry at infinity, is always three.
