Scale-invariant random geometry from mating of trees: a numerical study

Abstract

The search for scale-invariant random geometries is central to the Asymptotic Safety hypothesis for the Euclidean path integral in quantum gravity. In an attempt to uncover new universality classes of scale-invariant random geometries that go beyond surface topology, we explore a generalization of the mating of trees approach introduced by Duplantier, Miller and Sheffield. The latter provides an encoding of Liouville Quantum Gravity on the 2-sphere decorated by a certain random space-filling curve in terms of a two-dimensional correlated Brownian motion, that can be viewed as describing a pair of random trees. The random geometry of Liouville Quantum Gravity can be conveniently studied and simulated numerically by discretizing the mating of trees using the Mated-CRT maps of Gwynne, Miller and Sheffield. Considering higher-dimensional correlated Brownian motions, one is naturally led to a sequence of non-planar random graphs generalizing the Mated-CRT maps that may belong to new universality classes of scale-invariant random geometries. We develop a numerical method to efficiently simulate these random graphs and explore their possible scaling limits through distance measurements, allowing us in particular to estimate Hausdorff dimensions in the two- and three-dimensional setting. In the two-dimensional case these estimates accurately reproduce previous known analytic and numerical results, while in the three-dimensional case they provide a first window on a potential three-parameter family of new scale-invariant random geometries.

arXiv comment28 pages, 16 figures

References in corpus (19)

• Nonperturbative Evolution Equation for Quantum Gravity
• Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point
• Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change
• Quantum Gravity from Causal Dynamical Triangulations: A Review
• The topological structure of scaling limits of large planar maps
• Random walks in cones
• Fractal space-times under the microscope: A Renormalization Group view on Monte Carlo data
• World-Sheet Geometry and Baby Universes in 2-D Quantum Gravity
• Limit of normalized quadrangulations: The Brownian map
• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Two perspectives of the 2D unit area quantum sphere and their equivalence
• Characteristics of the new phase in CDT
• Spectral dimension of quantum geometries
• Upper bounds on Liouville first passage percolation and Watabiki's prediction
• On characterizing the Quantum Geometry underlying Asymptotic Safety
• The higher-order phase transition in toroidal CDT
• Spectral dimension on spatial hypersurfaces in causal set quantum gravity
• Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity
• Computation of Tight Enclosures for Laplacian Eigenvalues

Cited by in corpus (1)

• Phases of Tree-decorated Dynamical Triangulations in 3D