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 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 $T^{2/3}$.

arXiv comment44 pages, 1 figure

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• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Short time large deviations of the KPZ equation
• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
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• Lyapunov exponents of the half-line SHE
• KPZ equation with a small noise, deep upper tail and limit shape
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