Spacetime limit shapes of the KPZ equation in the upper tails

TL;DR

This paper characterizes the large deviation spacetime shapes of the KPZ equation's upper tails for multiple points at fixed time, revealing how noise concentrates in specific corridors across various time scales.

Contribution

It provides the first proof of the n-point large deviation principle and explicitly describes the spacetime limit shape and noise concentration corridors for the KPZ equation.

Findings

Established the n-point large deviation principle for KPZ upper tails.

Explicitly characterized the spacetime limit shape and noise corridors.

Validated the results across different time scaling regimes.

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.