Geodesic Lévy flights on Zoll surfaces

Yann Chaubet; Emanuel József Godfried; Leo Tzou

Geodesic Lévy flights on Zoll surfaces

Differential Geometry 2026-06-27 v1 Probability

Authors: Yann Chaubet , Emanuel József Godfried , Leo Tzou

Abstract

We study the mean first capture time of isotropic L\'evy flights on Zoll surfaces, namely the expected time for a geodesic L\'evy process to reach a shrinking geodesic ball. While the leading-order asymptotics are universal, we prove that the first correction term encodes subtle geometric information. More precisely, it is completely determined by the local singularity type of the conjugate locus, quantified by the degree of the conjugate point. This yields a hierarchy of asymptotic regimes governed by the L\'evy exponent.

Cite

@article{arxiv.2606.28744,
  title  = {Geodesic Lévy flights on Zoll surfaces},
  author = {Yann Chaubet and Emanuel József Godfried and Leo Tzou},
  journal= {arXiv preprint arXiv:2606.28744},
  year   = {2026}
}