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 ε in front of the noise and let ε→ 0. We prove that the one-point large deviation rate function has a 32 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 solution of the KPZ equation as ε→ 0. This confirms the physics prediction in Hartmann et al. (Phys Rev Res 1(3):032043, 2019), Kolokolov and Korshunov (Phys Rev B 75(14):140201, 2007, Phys Rev E 80(3):031107, 2009), Kamenev et al. (Phys Rev E 94(3):032108, 2016), Le Doussal et al. (Phys Rev Lett 117(7):070403, 2016) and Meerson et al. (Phys Rev Lett 116(7):070601, 2016).