The two-dimensional KPZ equation in the entire subcritical regime

Abstract

We consider the KPZ equation in space dimension 2 driven by spacetime white noise. We showed in previous work that if the noise is mollified in space on scale ε and its strength is scaled as β/√|log ε|, then a transition occurs with explicit critical point βc =√2π. Recently Chatterjee and Dunlap showed that the solution admits subsequential scaling limits as ε ↓ 0, for sufficiently small β. We prove here that the limit exists in the entire subcritical regime β ∈ (0,βc) and we identify it as the solution of an additive stochastic heat equation, establishing so-called Edwards-Wilkinson fluctuations. The same result holds for the directed polymer model in random environment in space dimension 2.