Simple derivation of the $(- \lambda H)^{5/2}$ tail for the 1D KPZ equation

TL;DR

This paper derives a simple, exact expression for the tail of the large deviations in the height distribution of the 1D KPZ equation, valid at all times, for various initial conditions, revealing a universal $(- ilde{s})^{5/2}$ tail.

Contribution

The paper provides a novel, straightforward derivation of the $(- ilde{s})^{5/2}$ tail for the KPZ height distribution's large deviations, extending previous short-time results to all times.

Findings

The tail of the large deviations follows a $(- ilde{s})^{5/2}$ form with a specific prefactor.

The tail expression is valid for both Brownian and droplet initial conditions.

The same tail appears in half-space KPZ with a different prefactor.

Abstract

We study the long-time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions for the Brownian and droplet initial conditions and present a simple derivation of the tail of the large deviations of the height on the negative side $\lambda H<0$. We show that for both initial conditions, the cumulative distribution functions take a large deviations form, with a tail for $- \tilde s \gg 1$ given by $-\log \mathbb{P}\left(\frac{H}{t}<\tilde{s}\right)=t^2 \frac{4 }{15 \pi} (-\tilde{s})^{5/2} $. This exact expression was already observed at small time for both initial conditions suggesting that these large deviations remain valid at all times. We present two methods to derive the result (i) long time estimate using a Fredholm determinant formula and (ii) the evaluation of the cumulants of a determinantal point process where the successive cumulants appear to give the successive orders of the large deviation rate function in the large $\tilde s$ expansion. An interpretation in terms of large deviations for trapped fermions at low temperature is also given. In addition, we perform a similar calculation for the KPZ equation in a half-space with a droplet initial condition, and show that the same tail as above arises, with the prefactor $\frac{4}{15\pi}$ replaced by $\frac{2}{15\pi}$. Finally, the arguments can be extended to show that this tail holds for all times. This is consistent with the fact that the same tail was obtained previously in the short time limit for the full-space problem.