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The present work is devoted to the study of the large time behaviour of acritical Brownian diffusion in two dimensions, whose drift is divergence-free,ergodic and given by the curl of the 2-dimensional Gaussian Free Field. Weprove 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 superdiffusivebehaviour has been predicted to occur for a wide variety of out-of-equilibriumsystems in the critical spatial dimension $d=2$. Examples include the diffusionof 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 thesecases 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 isrigorously established.

$\sqrt{\log t}$-superdiffusivity for a Brownian particle in the curl of the 2d GFF