  Motivated by the search for geometric observables in nonperturbative quantum
gravity, we define a notion of coarse-grained Ricci curvature. It is based on a
particular way of extracting the local Ricci curvature of a smooth Riemannian
manifold by comparing the distance between pairs of spheres with that of their
centres. The quantum Ricci curvature is designed for use on non-smooth and
discrete metric spaces, and to satisfy the key criteria of scalability and
computability. We test the prescription on a variety of regular and random
piecewise flat spaces, mostly in two dimensions. This enables us to quantify
its behaviour for short lattices distances and compare its large-scale
behaviour with that of constantly curved model spaces. On the triangulated
spaces considered, the quantum Ricci curvature has good averaging properties
and reproduces classical characteristics on scales large compared to the
discretization scale.
