November 2022 logt-Superdiffusivity for a Brownian particle in the curl of the 2D GFF
Giuseppe Cannizzaro, Levi Haunschmid-Sibitz, Fabio Toninelli
Author Affiliations +
Ann. Probab. 50(6): 2475-2498 (November 2022). DOI: 10.1214/22-AOP1589


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 (J. Stat. Phys. 147 (2012) 113–131), according to which the diffusion coefficient D(t) diverges as logt for t. Starting from the fundamental work by Alder and Wainwright (Phys. Rev. Lett. 18 (1967) 988–990), 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)logt, to the best of the authors’ knowledge, this is the first instance in which such precise asymptotics is rigorously established.

Funding Statement

G. C. gratefully acknowledges financial support via the EPSRC Grant EP/ S012524/1. F. T. gratefully acknowledges financial support of Agence Nationale de la Recherche via the ANR-15-CE40-0020-03 Grant LSD and of the Austria Science Fund (FWF), Project Number P 35428-N.


The authors would like to thank Bálint Tóth and Benedek Valkó for enlightening discussions, and the anonymous referee for a very careful reading and for comments that allowed us to improve the main statement and streamline some technical arguments.


Download Citation

Giuseppe Cannizzaro. Levi Haunschmid-Sibitz. Fabio Toninelli. "logt-Superdiffusivity for a Brownian particle in the curl of the 2D GFF." Ann. Probab. 50 (6) 2475 - 2498, November 2022.


Received: 1 June 2021; Revised: 1 May 2022; Published: November 2022
First available in Project Euclid: 23 October 2022

MathSciNet: MR4499841
zbMATH: 1502.82015
Digital Object Identifier: 10.1214/22-AOP1589

Primary: 60H10 , 82C27

Keywords: diffusion coefficient , Diffusion in random environment , Gaussian free field , Super-diffusivity

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 6 • November 2022
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