When the value L of the directed landscape at a point is sufficiently large, the geodesic from p to q is rigid and its location fluctuates of order around its expectation. We further show that at a midpoint of the geodesic, the location of the geodesic and the value of the directed landscape after appropriate scaling converge to two independent Gaussians.
The work was supported by the University of Kansas Start Up Grant, the University of Kansas New Faculty General Research Fund, Simons Collaboration Grant No. 637861, and NSF grant DMS-1953687.
We would like to thank Jinho Baik, Ivan Corwin, Duncan Dauvergne, and Bálint Virág for the comments and suggestions.
"When the geodesic becomes rigid in the directed landscape." Electron. Commun. Probab. 27 1 - 13, 2022. https://doi.org/10.1214/22-ECP484