The two-dimensional KPZ equation in the entire subcritical regime

Abstract

We consider the KPZ equation in space dimension $2$ driven by space-time white noise. We showed in previous work that if the noise is mollified in space on scale $\varepsilon $ and its strength is scaled as $\hat{\beta }/\sqrt{|\log \varepsilon |}$, then a transition occurs with explicit critical point $\hat{\beta }_{c}=\sqrt{2\pi }$. Recently Chatterjee and Dunlap showed that the solution admits subsequential scaling limits as $\varepsilon \downarrow 0$, for sufficiently small $\hat{\beta }$. We prove here that the limit exists in the entire subcritical regime $\hat{\beta }\in (0,\hat{\beta }_{c})$ 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 $2$.