Solving marginals of the LDP for the directed landscape

Abstract

We identify the explicit rate function for the upper-tail LDP of the parabolic Airy process and characterize the minimizing metrics, or equivalently 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, DDV), and the limit shape result gives the first proof of the limit shape in similar contexts beyond the case where the minimizer is piecewise linear. The starting point of our proof is the metric-level LDP for the directed landscape from DDV that reduces our work to solving a variational problem. Our proof demonstrates how the variational problem fits into the Jensen-Varadhan theory of hydrodynamic large deviations Jensen (2000), Varadhan (2004). The proof is PDE-based and uses geometric arguments, connecting the variational problem to weak solutions of Burgers' equation, and may be generalized to those models in the KPZ universality class that enjoy similar metric-level LDPs proven in DDV.