Geometry of semi-infinite geodesics in the Airy line ensemble

Source abstract

We study the geometry of semi-infinite geodesics in the Airy line ensemble. We establish sharp estimates for disjointness probabilities involving geodesics with nearby directions and endpoints in the top line of the Airy line ensemble. Moreover, we obtain a power law upper bound on the coalescence depth of two semi-infinite geodesics with same asymptotic direction and establish exponential concentration estimates for geodesic jump times more generally. We also establish pathwise Hölder $1/2^-$-continuity of Busemann profiles associated to semi-infinite geodesics with common asymptotic directions (in terms of the asymptotic direction). We characterize exceptional behaviour of semi-infinite geodesics through their Busemann profiles and establish geometric properties of the associated sets of non-coalescence and non-uniqueness. Finally, we determine the Hausdorff dimensions of collections of semi-infinite geodesics: collections with a fixed top-line endpoint and arbitrary positive asymptotic direction almost surely have Hausdorff dimension $1$, while collections with a fixed positive direction and arbitrary endpoint almost surely have Hausdorff dimension $3/2$.