Ricci curvature and quantum geometry

We describe a few elementary aspects of the circle of ideas associated with a quantum field theory (QFT) approach to Riemannian Geometry, a theme related to how Riemannian structures are generated out of the spectrum of (random or quantum) fluctuations around a background fiducial geometry. In such a scenario, Ricci curvature with its subtle connections to diffusion, optimal transport, Wasserestein geometry, and renormalization group, features prominently.