  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.
