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

TL;DR

This paper investigates the large-time behavior of a passive tracer influenced by a Gaussian free field in two dimensions, establishing that its mean-squared displacement grows like t times the square root of the logarithm of t, confirming physics predictions.

Contribution

It provides a rigorous proof of the asymptotic growth rate of the tracer's displacement, using stochastic homogenization techniques, and extends previous physics-based predictions to a mathematical setting.

Findings

Mean-squared displacement scales as t√ln t

Confirmed physics predictions with rigorous proof

Analyzed effective diffusivity with 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.