The Gaussian free-field as a stream function: Asymptotics of effective diffusivity in infra-red cut-off
Georgiana Chatzigeorgiou, Peter S. Morfe, Félix Otto, Lihan Wang · The Annals of Probability · 2025
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 lnt, as predicted in the physics literature. This improves the asymptotics recently established in recent work of Cannizzaro, Haunschmidt-Sibitz, and Toninelli (Ann. Probab. 50 (2022) 2475–2498), which uses mathematical-physics type analysis in Fock space. Our approach involves studying the effective diffusivity λL of the process with an infra-red cut-off at scale L, and is based on techniques from stochastic homogenization.