Combinatorial quantum gravity and emergent 3D quantum behaviour

Abstract: We review combinatorial quantum gravity, an approach which combines Einstein's idea of dynamical geometry with Wheeler's "it from bit" hypothesis in a model of dynamical graphs governed by the coarse Ollivier-Ricci curvature. This drives a continuous phase transition from a random to a geometric phase, due to a condensation of loops on the graph. In the 2D case, the geometric phase describes negative-curvature surfaces with two inversely related scales, an ultraviolet (UV) Planck length and an infrared (IR) radius of curvature. Below the Planck scale the random bit character survives: chunks of random bits of the Planck size describe matter particles of excitation energy given by their excess curvature. Between the Planck length and the curvature radius, the surface is smooth, with spectral and Hausdorff dimension 2; at scales larger than the curvature radius, particles see the surface as an effective Lorentzian de Sitter surface, the spectral dimension becomes 3 and the effective slow dynamics of particles, as seen by co-moving observers, emerges as quantum mechanics in Euclidean 3D space. Since the 3D distances are inherited from the underlying 2D de Sitter surface, we obtain curved trajectories around massive particles also in 3D, representing the large-scale gravity interactions. We shall thus propose that this 2D model describes a generic holographic screen relevant for real quantum gravity.