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

Abstract

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.

25 pages; In v2, we added Section 9, where we establish real-time super-diffusivity results for the SDE without infra-red cut-off; In v3, we incorporated changes suggested by anonymous viewers (accepted version)