  We analyze the large-time asymptotics of a passive tracer with drift equal to
the curl of the Gaussian free field in two dimensions with ultra-violet cut-off
at scale unity. We prove that the mean-squared displacement scales like $t
\sqrt{\ln t}$, as predicted in the physics literature and recently almost
proved by the work of Cannizzaro, Haunschmidt-Sibitz, and Toninelli (2022),
which uses mathematical-physics type analysis in Fock space. Our approach
involves studying the effective diffusivity $\lambda_{L}$ of the process with
an infra-red cut-off at scale $L$, and is based on techniques from stochastic
homogenization.
