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 diverges as for . 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 . 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 , to the best of the authors’ knowledge, this is the first instance in which such precise asymptotics is rigorously established.
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.
"-Superdiffusivity for a Brownian particle in the curl of the 2D GFF." Ann. Probab. 50 (6) 2475 - 2498, November 2022. https://doi.org/10.1214/22-AOP1589