Solving marginals of the LDP for the directed landscape

TL;DR

This paper establishes the upper-tail Large Deviation Principle for the parabolic Airy process and characterizes the limit shape of the directed landscape under this condition, advancing understanding of large deviations in stochastic growth models.

Contribution

It proves the upper-tail LDP for the parabolic Airy process, confirming a conjecture and connecting variational problems to PDE solutions, with potential generalizations to KPZ fixed point.

Findings

Proved the upper-tail LDP for the parabolic Airy process.

Characterized the limit shape of the directed landscape under upper-tail conditioning.

Connected variational problems to weak solutions of Burgers' equation.

Abstract

We prove the upper-tail Large Deviation Principle (LDP) for the parabolic Airy process and characterize the limit shape of the directed landscape under the upper-tail conditioning. The LDP result answers Conjecture 10.1 in Das, Dauvergne, and Vir\'{a}g (2024). The starting point of our proof is the metric-level LDP for the directed landscape from Das, Dauvergne, and Vir\'{a}g (2024) that reduces our work to solving a variational problem. Our proof is PDE-based and uses geometric arguments, connecting the variational problem to the weak solutions of Burgers' equation. Further, our method may generalize to the setting of the upper-tail LDP for the KPZ fixed point under the multi-wedge initial data, and we prove a decomposition result in this direction.