Lower tail of the KPZ equation

Abstract

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