Title: Solving marginals of the LDP for the directed landscape.
Abstract: Recently, Das, Dauvergne, and Virág proved the upper-tail Large Deviation Principle (LDP) for the directed landscape, with a rate function written in terms of (directed) metrics. In this talk, we explain how one can solve the corresponding metric-level variational problem to obtain some marginal LDPs. Specifically, we prove the upper-tail LDP for the parabolic Airy process and characterize the limit shape of the directed landscape under the upper-tail conditioning. Our method is PDE-based and uses geometric arguments, connecting the variational problem to the weak solutions of Burgers' equation. Joint work with Sayan Das.