  Recent results obtained within a non-perturbative approach to quantum gravity
based on the method of four-dimensional Causal Dynamical Triangulations are
described. The phase diagram of the model consists of three phases. In the
physically most interesting phase, the time-translational symmetry is
spontaneously broken. Calculations of expectation values required introducing
procedures taking into account the inhomogeneity of configurations. It was
shown that the dynamically emerged four-dimensional background geometry
corresponds to a Euclidean de Sitter space and reveals no fractality at large
distances. Measurements of the covariance matrix of scale factor fluctuations
allowed to reconstruct the effective action, which remained in agreement with
the discrete minisuperspace action. Values of the Hausdorff dimension and
spectral dimension of three-dimensional spatial slices suggest their fractal
nature, which was confirmed by a direct analysis of triangulation structure.
The Monte Carlo algorithm used to obtain presented results is described.
