One-point large deviations of the directed landscape geodesic

TL;DR

This work resolves conjectures about tail probabilities and the geometry of directed landscape geodesics by establishing a precise large-deviation principle for the rightmost geodesic through a fixed point. It translates the problem into a rate-function minimization over directed metrics with planted measures, proving the existence and uniqueness of a minimizer (F,ρ) with a canonical structure: a linear segment up to a merge time and a parabolic tail thereafter, coupled with a constant density on [0,t_B]. The authors derive an explicit minimal rate, $I_{ ext{min}}=\frac{8}{3a^2(3-\sqrt{8a})^2}$, and provide the exact geodesic shape and density for $a<\tfrac{1}{2}$, together with convergence results linking rescaled directed landscapes to these rate-function objects. These results sharpen our understanding of KPZ universality by giving concrete large-deviation asymptotics for geodesic deviation and establishing a rigorous optimization framework for geodesic conditioning.

Abstract

The directed landscape, the central object in the Kardar-Parisi-Zhang universality class, is shown to be the scaling limit of various models by Dauvergne and Virág (2022) and Dauvergne, Ortmann and Virág (2018). In his study of geodesics in upper tail deviations of the directed landscape, Liu (2022) put forward a conjecture about the rate of the lowest rate metric under which a geodesic between two points passes through a particular point between them. Das, Dauvergne and Virág (2024) disproved his conjecture, and made a conjecture of their own. This paper disproves that conjecture and puts the question to rest with an answer and a proof.

Figures (2)

• Figure 1: $0 < a \leq \frac{1}{8}$
• Figure 2: $\frac{1}{8} < a \leq \frac{1}{2}$

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof