Uniformity of extremal behaviour along geodesics in Liouville quantum gravity

Abstract

In random geometry, geodesics often have a tendency to coalesce together and traverse regions highly singular relative to typical environments. In this paper, working with the model of Liouville quantum gravity, we develop a technique which yields zero-one laws for many extremal statistics measured along interior segments of the geodesic. Namely, working with statistics such as the Euclidean dimension, optimal Hölder continuity exponents with respect to the Euclidean metric and the minimal/maximal encountered thickness for the underlying GFF, we obtain zero-one laws for the above for segments in the bulk of geodesics, thereby upgrading the results of Gwynne-Pfeffer-Sheffield '22. The primary technique used, developed in Bhatia-Kavvadias '25, is to lay down a large family of small-scale "typical" and well-behaved geodesics along interior segments of a long geodesic.