$\sqrt{\log t}$-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 [B. T\'oth, B. Valk\'o, J. Stat. Phys., 2012], according to which the diffusion coefficient $D(t)$ diverges as $\sqrt{\log t}$ for $t\to\infty$. Starting from the fundamental work by Alder and Wainwright [B. Alder, T. Wainright, Phys. Rev. Lett. 1967], logarithmically superdiffusive behaviour has been predicted to occur for a wide variety of out-of-equilibrium systems in the critical spatial dimension $d=2$. 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 $D(t)\sim\sqrt{\log t}$, to the best of the authors' knowledge, this is the first instance in which such precise asymptotics is rigorously established.