Kolloquium Wilhelm Killing

The interplay between probability theory and number theory has a rich history of producing deep results and conjectures. Important instances are the works of Erdös, Kac, Selberg, Montgomery, Soundararajan and Granville, to name a few. This talk will review recent results in this spirit where the insights of probability, of branching processes in particular, have led to a better understanding of large values of the Riemann zeta function on the critical line.

I will discuss the large-scale/long-time behavior of a Brownian particle advected by a random, incompressible (divergence-free) vector field. If this vector field has correlations which decay with a critical exponent (which happens to be -2), then the behavior of the particle is superdiffusive instead of diffusive. In particular, after time $t$, the particle will be on average a distance of $t^{\frac12} (\log t)^{\frac14}$ from its starting point. This phenomenon was explained in the physics literature in the late 1980s as a build-up of diffusivity across many length scales, using renormalization group arguments. I will discuss recent mathematical innovations based on analytic ideas that allow us to make these renormalization group arguments rigorous. (This is joint work with A. Bou-Rabee and T. Kuusi.)