In Riemannian geometry, geodesics are curves which locally minimize lengths. In general, it is a difficult and interesting question to determine which geodesics of a manifold are in fact globally minimizing. In settings of non‐positive curvature (e.g., hyperbolic space), the Cartan‐Hadamard theorem says that all geodesics are minimizing, so the presence of positive curvature (e.g., sphere) is necessary to destabilize this minimization property. With Janek Wehr, we have used the point‐of‐view of the particle technique to show that for random perturbations of 2‐dimensional Euclidean space, enough positive curvature arises to destabilize "generic" geodesics. I will present this work, as well as discuss the extension to the more general setting of symmetric random geometries. No background in geometry or probability will be required for this talk, and it will be accessible to graduate students.