  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 establish 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. The same approach also applies to the Bacry-Emery curvature
condition CD$(0,\infty)$, thereby settling a recent conjecture of Cushing, Liu
and Peyerimhoff (2019).
