Cannizzaro, Giuseppe; Haunschmid-Sibitz, Levi; Toninelli, Fabio \(\sqrt{\log t}\)-superdiffusivity for a Brownian particle in the curl of the 2D GFF. (English) Zbl 1502.82015 Ann. Probab. 50, No. 6, 2475-2498 (2022). Summary: 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. Tóth and B. Valkó, J. Stat. Phys. 147, No. 1, 113–131 (2012; Zbl 1245.82092)], according to which the diffusion coefficient \(D(t)\) diverges as \(\sqrt{\log t}\) for \(t\to \infty\). Starting from the fundamental work by B. J. Alder and T. E. Wainwright [“Velocity autocorrelations for hard spheres”, Phys. Rev. Lett. 18, No. 23, 988–990 (1967; doi:10.1103/PhysRevLett.18.988)], 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. 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