# Large deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $\lambda H<0$

## TL;DR

This paper derives exact large deviation functions for rare height fluctuations in the 1+1D KPZ equation at long times, revealing a crossover between Tracy-Widom and different tail behaviors, supported by numerical analysis.

## Contribution

It provides the first exact long-time large deviation function for height fluctuations in the KPZ equation for $ ext{sign}( ext{nonlinearity})<0$, revealing a crossover in tail behavior.

## Key findings

- Identified a crossover from Tracy-Widom to a different tail at large deviations.
- Derived exact large deviation functions for $ ext{sign}( ext{nonlinearity})<0$.
- Supported analytical results with numerical evaluations.

## Abstract

We study atypically large fluctuations of height $H$ in the 1+1-dimensional Kardar-Parisi-Zhang (KPZ) equation at long times $t$, when starting from a "droplet" initial condition. We derive exact large deviation function of height for $\lambda H<0$, where $\lambda$ is the nonlinearity coefficient of the KPZ equation. This large deviation function describes a crossover from the Tracy-Widom distribution tail at small $|H|/t$, which scales as $|H|^3/t$, to a different tail at large $|H|/t$, which scales as $|H|^{5/2}/t^{1/2}$. The latter tail exists at all times $t>0$. It was previously obtained in the framework of the optimal fluctuation method. It was also obtained at short times from exact representation of the complete height statistics. The crossover between the two tails, at long times, occurs at $|H|\sim t$ as previously conjectured. Our analytical findings are supported by numerical evaluations using exact representation of the complete height statistics.