KPZ equation with a small noise, deep upper tail and limit shape

TL;DR

This paper analyzes the KPZ equation with small noise, establishing the deep upper tail large deviation rate function's power law and deriving a limit shape as noise vanishes, confirming several physics predictions.

Contribution

It proves the deep upper tail large deviation rate function's power law and derives the limit shape of KPZ under weak noise, confirming prior physics-based predictions.

Findings

Deep upper tail rate function follows a 3/2 power law.

Limit shape of KPZ confirmed as noise parameter tends to zero.

Results align with previous physics predictions.

Abstract

In this paper, we consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter $\sqrt{\varepsilon}$ in front of the noise and let $\varepsilon \to 0$. We prove that the one-point large deviation rate function has a $\frac{3}{2}$ power law in the deep upper tail. Furthermore, by forcing the value of the KPZ equation at a point to be very large, we prove a limit shape of the KPZ equation as $\varepsilon \to 0$. This confirms the physics prediction in Kolokolov and Korshunov (2007), Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Kamenev, Meerson, and Sasorov (2016), and Le Doussal, Majumdar, Rosso, and Schehr (2016).