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 $\epsilon$ and its strength is scaled as $\hat{\beta }/\sqrt{|\log \epsilon |}$, 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 $\epsilon \downarrow 0$, 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.

42 pages, 2 figures. Final version, to appear in AOP