$\sqrt{\log t}$-superdiffusivity for a Brownian particle in the curl of the 2d GFF

TL;DR

This paper rigorously proves that a 2D Brownian particle influenced by the curl of the Gaussian Free Field exhibits superdiffusive behavior, with its diffusion coefficient diverging as the square root of the logarithm of time.

Contribution

The authors establish the first rigorous proof that the diffusion coefficient diverges as \\sqrt{\\log t} for a Brownian particle driven by a divergence-free Gaussian Free Field in two dimensions.

Findings

Diffusion coefficient diverges as \\sqrt{\\log t} as t \\to \\infty

Confirms a long-standing conjecture in the critical 2D case

Provides rigorous mathematical foundation for superdiffusive behavior in 2D systems

Abstract

The present work is devoted to the study of the large time behaviour of a critical Brownian diffusion in two dimensions, whose drift is divergence-free, ergodic and given by the curl of the 2-dimensional Gaussian Free Field. We prove the conjecture, made in [B. T\'oth, B. Valk\'o, J. Stat. Phys., 2012], according to which the diffusion coefficient $D(t)$ diverges as $\sqrt{\log t}$ for $t\to\infty$. Starting from the fundamental work by Alder and Wainwright [B. Alder, T. Wainright, Phys. Rev. Lett. 1967], logarithmically superdiffusive behaviour has been predicted to occur for a wide variety of out-of-equilibrium systems in the critical spatial dimension $d=2$. Examples include the diffusion of a tracer particle in a fluid, self-repelling polymers and random walks, Brownian particles in divergence-free random environments, and, more recently, the 2-dimensional critical Anisotropic KPZ equation. Even if in all of these cases it is expected that $D(t)\sim\sqrt{\log t}$, to the best of the authors' knowledge, this is the first instance in which such precise asymptotics is rigorously established.