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 (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.