Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape

TL;DR

This paper investigates the fractal properties of geodesic paths in the directed landscape, showing that sets of endpoints with disjoint geodesics have Hausdorff dimension one-half, revealing intricate geometric structures.

Contribution

It establishes the Hausdorff dimension of sets of disjoint geodesics in the directed landscape, providing new insights into their fractal geometry and extending previous tail estimates.

Findings

Hausdorff dimension of disjoint geodesic endpoint sets is one-half.

Sets of geodesics coalescing only at the final time have Hausdorff dimension one-half.

Extension of tail estimates for disjoint geodesics from pre-limiting models to the directed landscape.

Abstract

Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work arXiv:1812.00309 of Dauvergne, Ortmann, and Vir\'ag, this object was constructed and shown to be the limit after parabolic correction of one such model: Brownian last passage percolation. This limit object, called the directed landscape, admits geodesic paths between any two space-time points $(x,s)$ and $(y,t)$ with $s<t$. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations $x_1$ and $x_2$, and consider geodesics traveling $(x_1,0)\to (y,1)$ and $(x_2,0)\to (y,1)$. We prove that the set of $y\in\mathbb{R}$ for which these geodesics coalesce only at time $1$ has Hausdorff dimension one-half. Second, we consider endpoints $(x,0)$ and $(y,1)$ between which there exist two geodesics intersecting only at times $0$ and $1$. We prove that the set of such $(x,y)\in\mathbb{R}^2$ also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in arXiv:1904.01717; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in arXiv:1709.04110.