logt-Superdiffusivity for a Brownian particle in the curl of the 2D GFF

Abstract

The present work is devoted to the study of the large time behaviour of a critical Brownian diffusion in two dimensions, whose drift is divergence-free, ergodic and given by the curl of the 2-dimensional Gaussian free field. We prove the conjecture, made in (J. Stat. Phys. 147 (2012) 113–131), according to which the diffusion coefficient $\mathit{D}(\mathit{t})$ diverges as $\sqrt{log\mathit{t}}$ for $\mathit{t}\to \infty$. Starting from the fundamental work by Alder and Wainwright (Phys. Rev. Lett. 18 (1967) 988–990), logarithmically superdiffusive behaviour has been predicted to occur for a wide variety of out-of-equilibrium systems in the critical spatial dimension $\mathit{d}=2$. Examples include the diffusion of a tracer particle in a fluid, self-repelling polymers and random walks, Brownian particles in divergence-free random environments and, more recently, the 2-dimensional critical Anisotropic KPZ equation. Even if in all of these cases it is expected that $\mathit{D}(\mathit{t})\sim \sqrt{log\mathit{t}}$, to the best of the authors’ knowledge, this is the first instance in which such precise asymptotics is rigorously established.