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

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.