Final Activity Report Summary - ENRAGE (European Network on Random Geometry) Progress in sciences is driven by the urge to push the limits of our understanding of the physical world. The unprecedented advances of the last century led to a collective search, by theoretical physicists, for the most fundamental building blocks of space, time and matter, as well as for a unified description of their interactions. In our effort to formulate a quantum theory of physics at the most extreme scales, there is mounting evidence that special, so-called non-perturbative, methods are necessary. These must take into account that space-time at the Planck scale is not well approximated by the fixed flat Minkowski space which provides the setting for standard quantum field theory at much lower energies. The geometry of space-time itself takes part dynamically and must be treated in a manner consistent with both quantum theory and general relativity. The research undertaken by the ENRAGE collaboration made significant strides towards constructing a non-perturbative, background-independent theory of quantum gravity using a combination of analytic and numerical tools from the theory of discrete random geometries and based on the so-called causal dynamical triangulations. Using these methods, it was possible for the first time to derive a classical, four-dimensional universe, starting from the microscopic quantum dynamics of sub-Planckian constituents of space-time alone. This dynamically generated universe had the shape of a de Sitter space, a well-known solution to the classical Einstein equations in the presence of a cosmological constant, i.e. dark energy. The same powerful methods, involving random geometry and random matrices, were used by network researchers to address an amazing variety of problems in various dimensions, such as one-dimensional random networks, graphs and trees, two-dimensional surfaces, e.g. string world sheets and membranes, and higher-dimensional geometries as they appeared, for example, in quantum gravity. We also successfully applied them in statistical mechanics and field theory, lattice quantum chromo-dynamics, pure mathematics, i.e. combinatorics and enumeration problems, econophysics and selected aspects of biology. A key towards the success was the smooth interplay between analytical methods on the one hand and computational Monte Carlo simulations on the other.
