Properties of dynamical fractal geometries in the model of Causal Dynamical Triangulations

We investigate the geometry of a quantum universe with the topology of the four-torus. The study of non-contractible geodesic loops reveals that a typical quantum geometry consists of a small semi-classical toroidal bulk part, dressed with many outgrowths, which contain most of the four-volume and which have almost spherical topologies, but nevertheless are quite fractal.Comment: 16 pages, 16 figure

Cosmic voids and filaments from quantum gravity

Using computer simulations we study the geometry of a typical quantum universe, i.e. the geometry one might expect before a possible period of inflation. We display it using coordinates defined by means of four classical scalar fields satisfying the Laplace equation with non-trivial boundary conditions. The field configurations reveal cosmic web structures surprisingly similar to the ones observed in the present-day Universe. Inflation might make these structures relevant for our Universe.Comment: 4 pages, 2 figure

Signal from noise retrieval from one and two-point Green's function - comparison

We compare two methods of eigen-inference from large sets of data, based on the analysis of one-point and two-point Green's functions, respectively. Our analysis points at the superiority of eigen-inference based on one-point Green's function. First, the applied by us method based on Pad?e approximants is orders of magnitude faster comparing to the eigen-inference based on uctuations (two-point Green's functions). Second, we have identified the source of potential instability of the two-point Green's function method, as arising from the spurious zero and negative modes of the estimator for a variance operator of the certain multidimensional Gaussian distribution, inherent for the two-point Green's function eigen-inference method. Third, we have presented the cases of eigen-inference based on negative spectral moments, for strictly positive spectra. Finally, we have compared the cases of eigen-inference of real-valued and complex-valued correlated Wishart distributions, reinforcing our conclusions on an advantage of the one-point Green's function method.Comment: 14 pages, 8 figures, 3 table

A unified fluctuation formula for one-cut β\beta-ensembles of random matrices

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $\beta$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $\beta$-ensemble. As particular cases of the main result we consider the classical $\beta$-Gaussian, $\beta$-Wishart and $\beta$-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian Unitary Ensemble and the enumeration of planar maps.Comment: 16 pages, 4 figures, 3 tables. Revised version where references have been added and typos correcte

Matter-driven change of spacetime topology

Using Monte-Carlo computer simulations, we study the impact of matter fields on the geometry of a typical quantum universe in the CDT model of lattice quantum gravity. The quantum universe has the size of a few Planck lengths and the spatial topology of a three-torus. The matter fields are multi-component scalar fields taking values in a torus with circumference $\delta$ in each spatial direction, which acts as a new parameter in the CDT model. Changing $\delta$, we observe a phase transition caused by the scalar field. This discovery may have important consequences for quantum universes with non-trivial topology, since the phase transition can change the topology to a simply connected one.Comment: 5 pages, 5 figure