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Geometrical properties of spacetime are difficult to study in nonperturbativeapproaches to quantum gravity like Causal Dynamical Triangulations (CDT), whereone uses simplicial manifolds to define the gravitational path integral,instead of Riemannian manifolds. In particular, in CDT one only relies on twomathematical tools, a distance measure and a volume measure. In this paper, wedefine a notion of scalar curvature, for metric spaces endowed with a volumemeasure or a random walk, without assuming nor using notions of tensorcalculus. Furthermore, we directly define the Ricci scalar, without the need ofdefining and computing the Riemann or the Ricci tensor a priori. For this, wemake use of quantities, like the surface of a geodesic sphere, or the returnprobability of scalar diffusion processes, that can be computed in these metricspaces, as in a Riemannian manifold, where they receive scalar curvaturecontributions. Our definitions recover the classical results of scalarcurvature when the sets are Riemannian manifolds. We propose seven methods tocompute the scalar curvature in these spaces, and we compare their features innatural implementations in discrete spaces. The defined generalized scalarcurvatures are easily implemented on discrete spaces, like graphs. We presentthe results of our definitions on random triangulations of a 2D sphere andplane. Additionally, we show the results of our generalized scalar curvatureson the quantum geometries of 2D CDT, where we find that all our definitionsindicate a flat ground state of the gravitational path integral.

Scalar curvature for metric spaces: Defining curvature for Quantum Gravity without coordinates