Abstract

We consider an advection-diffusion (or “passive scalar”) equation with a divergence-free vector field, which is a stationary random field exhibiting “critical” correlations. Predictions from physicists in the 80s state that, almost surely, this equation should behave like a heat equation at large scales, but with a diffusivity that diverges as the square root of the log of the scale. In joint work with Ahmed Bou-Rabee and Tuomo Kuusi, we give a rigorous proof of this prediction using an iterative quantitative homogenization procedure, which is a way of formalizing a renormalization group argument. The idea is to consider a scale decomposition of the vector field, and coarse-grain the equation, scale-by-scale. The random swirls of the vector field at each scale enhance the effective diffusivity. As we zoom out, we obtain an ODE for the effective diffusivity as a function of the scale, allow us to deduce that it diverges at the predicted rate. Meanwhile, new coarse-graining arguments allow us to rigorously integrate out the smaller scales in the equation and prove the result.