Spacetime Limit Shapes of the KPZ Equation in the Upper Tails

Abstract

We consider the n-point, fixed-time large deviations of the KPZ equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a wide range of scaling regimes that allows time to be short, unit-order, and long. We prove the n-point large deviation principle and characterize, with proof, the corresponding spacetime limit shape. Our proof is based on the results—from the companion paper (Tsai in High moments of the SHE in the clustering regimes, 2023)—on moments of the stochastic heat equation and utilizes ideas coming from a tree decomposition. Behind our proof lies the phenomenon where the major contribution of the noise concentrates around certain corridors in spacetime, and we explicitly describe the corridors.