Local and Global Geometry of Directed Last-Passage Percolation Models

Abstract

This thesis focuses on the study of exactly-solvable last-passage percolation (LPP) models, which are central objects in the Kardar-Parisi-Zhang universality class. Historically, most studies of these models are through algebraic formulae in integrable probability, bearing the disadvantage of being less informative and unrobust. In this thesis, we develop techniques in the direction of combining formulae and geometric arguments, and use them to solve two problems in the LPP models. For the first part, we demonstrate the convergence of the empirical distribution of local environments along geodesics, for the explicit model of exponential LPP. We also provide explicit formulae for the limiting distributions.These results essentially enable one to compute all local statistics along the geodesic. We also show convergence of the distribution of the environment around a typical point, and the local environments are homogeneous along geodesics. Our proofs make extensive use of a correspondence with TASEP as seen from a single second-class particle, for which we prove new results concerning ergodicity and convergence to equilibrium. For the second part, we study fractal geometry of the directed landscape, a continuous LPP model constructed recently in Dauvergne-Ortmann-Virag, and has since been shown to be the universal scaling limit of various exactly-solvable LPP models.We study the difference of passage times from two fixed points, introduced in Basu-Ganguly-Hammond. Owing to geodesic geometry, it turns out that the difference process is almost surely locally constant. The set of non-constancy is connected to disjointness of geodesics and inherits remarkable fractal properties. We introduce several new ideas and show that the set of non-constancy of the 2D difference process and the 1D temporal process have Hausdorff dimensions 5/3 and 2/3 respectively. A particularly crucial ingredient in our analysis is the novel construction of a local time process for the geodesic akin to Brownian local time, supported on the ``zero set'' of the geodesic, and we also show that such ``zero set'' has Hausdorff dimension 1/3.