I will discuss the large-scale/ long-time behavior of a Brownian particle in a random, incompressible vector field (or equivalently the solution of the corresponding "passive scalar" PDE). If this vector field has correlations which decay sufficiently slowly, then the behavior of the particle is superdiffusive instead of diffusive. In particular, after time $t$, the particle will typically be a much larger distance than $\sqrt{t}$ from the origin. This phenomenon was explained in the physics literature in the late 1980s, using heuristic renormalization group arguments, as the divergence of effective diffusivity as we zoom out to large length scales. I will discuss recent results that use analytic/PDE methods to make some of these renormalization group arguments rigorous. This is based on a new notion of "coarse-grained ellipticity" which may be of independent interest. (This is based on joint works with A. Bou-Rabee and T. Kuusi.)