  We introduce an ensemble of infinite causal triangulations, called the
uniform infinite causal triangulation, and show that it is equivalent to an
ensemble of infinite trees, the uniform infinite planar tree. It is proved that
in both cases the Hausdorff dimension almost surely equals 2. The infinite
causal triangulations are shown to be almost surely recurrent or, equivalently,
their spectral dimension is almost surely less than or equal to 2. We also
establish that for certain reduced versions of the infinite causal
triangulations the spectral dimension equals 2 both for the ensemble average
and almost surely. The triangulation ensemble we consider is equivalent to the
causal dynamical triangulation model of two-dimensional quantum gravity and
therefore our results apply to that model.
