This contribution is not intended as a review but, by suggestion of the editors, as a glimpse ahead into the realm of dually weighted tensor models for quantum gravity. This class of models allows one to consider a wider class of quantum gravity models, in particular one can formulate state sum models of spacetime with an intrinsic notion of foliation. The simplest one of these models is the one proposed by Benedetti and Henson [1], which is a matrix model formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this paper we apply the Functional Renormalization Group Equation (FRGE) to the Benedetti-Henson model with the purpose of investigating the possible continuum limits of this class of models. Possible continuum limits appear in this FRGE approach as fixed points of the renormalization group flow where the size of the matrix acts as the renormalization scale. Considering very small truncations, we find fixed points that are compatible with analytically known results for CDT in two dimensions. By studying the scheme dependence of our results we find that precision results require larger truncations than the ones considered in the present work. We conclude that our work suggests that the FRGE is a useful exploratory tool for dually weighted matrix models. We thus expect that the FRGE will be a useful exploratory tool for the investigation of dually weighted tensor models for CDT in higher dimensions.

In this paper, we investigate a critical behavior of JT gravity, a model of two-dimensional quantum gravity on constant negatively curved spacetimes. Our approach involves using techniques from random maps to investigate the generating function of Weil-Petersson volumes, which count random hyperbolic surfaces with defects. The defects are weighted geodesic boundaries, and criticality is reached by tuning the weights to the regime where macroscopic holes emerge in the hyperbolic surface, namely non-generic criticality. We analyze the impact of this critical regime on some universal features, such as its density of states. We present a family of models that interpolates between systems with ρ0(E) ~ $$ \sqrt{E-{E}_0} $$ E − E 0 and ρ0(E) ~ (E − E0)3/2, which are commonly found in models of JT gravity coupled to dynamical end-of-the-world and FZZT branes, and give a precise definition of what this phase transition means from the random geometry point of view.

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 scaleinvariant 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 twodimensional 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 nonplanar 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.