  We present a new regularisation of Euclidean Einstein gravity in terms of
(sequences of) graphs. In particular, we define a discrete Einstein-Hilbert
action that converges to its manifold counterpart on sufficiently dense random
geometric graphs (more generally on any sequence of graphs that converges to
the manifold in the sense of Gromov-Hausdorff). Our construction relies
crucially on the Ollivier curvature of optimal transport theory. Our methods
also allow us to define an analogous discrete action for Klein-Gordon fields.
These results may be taken as the basis for a combinatorial approach to quantum
gravity where we seek to generate graphs that approximate manifolds as
metric-measure structures.
