  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.
