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

The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off