√logt-SUPERDIFFUSIVITY FOR A BROWNIAN PARTICLE IN THE CURL OF THE 2D GFF

• 成果类型: Article
• 署名作者: Cannizzaro, Giuseppe; Haunschmid-Sibitz, Levi; Toninelli, Fabio
• 署名单位: University of Warwick; Technische Universitat Wien
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1589
• 发表日期: 2022
• 页码: 2475-2498
• 关键词: CENTRAL-LIMIT-THEOREM; random-walks; BEHAVIOR

摘要

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 root/log t for t ->infinity. Start-ing from the fundamental work by Alder and Wainwright (Phys. Rev. Lett. 18 (1967) 988-990), logarithmically superdiffusive behaviour has been pre-dicted 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) similar to root log t, to the best of the authors' knowledge, this is the first instance in which such precise asymptotics is rigorously established.