Sparse expanders have negative Ollivier-Ricci curvature

We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To prove this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible "at infinity". We then transfer this result to finite graphs via local weak convergence and a relative compactness argument. We believe that this "local weak limit" approach to mixing properties of Markov chains will have many other applications.