  The physical phase of Causal Dynamical Triangulations (CDT) is known to be
described by an effective, one-dimensional action in which three-volumes of the
underlying foliation of the full CDT play a role of the sole degrees of
freedom. Here we map this effective description onto a statistical-physics
model of particles distributed on 1d lattice, with site occupation numbers
corresponding to the three-volumes. We identify the emergence of the quantum
de-Sitter universe observed in CDT with the condensation transition known from
similar statistical models. Our model correctly reproduces the shape of the
quantum universe and allows us to analytically determine quantum corrections to
the size of the universe. We also investigate the phase structure of the model
and show that it reproduces all three phases observed in computer simulations
of CDT. In addition, we predict that two other phases may exists, depending on
the exact form of the discretised effective action and boundary conditions. We
calculate various quantities such as the distribution of three-volumes in our
model and discuss how they can be compared with CDT.
