$\sqrt{\log t}$ -二维GFF旋度中布朗粒子的超扩散系数
$\sqrt{\log t}$-superdiffusivity for a Brownian particle in the curl of the 2d GFF
摘要：
本文研究了二维临界布朗扩散的大时间行为，其漂移是无发散的、遍历的，并由二维高斯自由场的旋度给出.我们证明了[B]中的猜想。托特湾Valko，J. Stat.物理、2012]，据此扩散系数$D(t)$发散为对于$t\to\infty$的$\sqrt{\log t}$。从Alder和Wainwright的基础工作出发[B. Alder，T. Wainright，Phys. Rev. Lett. 1967年]，热力学超扩散行为已被预测发生在各种各样的临界空间维度$d=2$的平衡系统。例子包括示踪粒子在流体中的扩散，自排斥聚合物和随机游动，无发散随机环境中的布朗粒子，以及最近的二维临界各向异性KPZ方程。即使在所有这些情况下，预计$D(t)\sim\sqrt{\log t}$，尽作者所知，这是第一个例子，在这种精确的渐近严格建立。
英文摘要：
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 [B. Tóth, B. Valkó, 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.
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