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

In this paper, we consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter $\sqrt{\epsilon }$ in front of the noise and let $\epsilon \to 0$. 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 $\epsilon \to 0$. 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).