Lower tail of the KPZ equation

TL;DR

This paper establishes the first tight bounds on the lower tail probability of the KPZ equation with narrow wedge initial data, revealing a crossover in decay behavior depending on the tail depth and time scale.

Contribution

It provides the first rigorous bounds on the lower tail of the KPZ equation, highlighting a transition in decay exponents for different tail depths.

Findings

Crossover between decay exponents 5/2 and 3 depending on tail depth

Bounds valid for all sufficiently large times T

Explicit pre-factors for the tail decay rates

Abstract

We provide the first tight bounds on the lower tail probability of the one point distribution of the KPZ equation with narrow wedge initial data. Our bounds hold for all sufficiently large times $T$ and demonstrates a crossover between super-exponential decay with exponent $5/2$ (and leading pre-factor $\frac{4}{15\pi} T^{1/3}$) for tail depth greater than $T^{2/3}$, and exponent $3$ (with leading pre-factor $\frac{1}{12}$) for tail depth less than $T^{2/3}$.