Yilin Wang

We study the large deviations as $k$ goes to 0 of $n$-chordal $\text{SLE}_k$, a set of $n$ simple random curves connecting $2n$ boundary points of a simply connected domain, modeling the interfaces in 2D critical statistical mechanics models with alternating boundary conditions. The large deviation rate function is described using the Loewner potential that we introduced, which depends on the boundary data and the curves.

We show that the potential minimizers for a given boundary data are the real locus of rational functions. This connection provides a new proof of the Shapiro conjecture on the classification of rational functions with real critical points, first proved by Eremenko and Gabrielov. Moreover, the potential can be expressed in terms of determinants of Laplacians, and the minimal potential satisfies the semiclassical limit of PDEs arising from the Belavin-Polyakov-Zamolodchikov equations in conformal field theory. This is joint work with Eveliina Peltola (Bonn, Aalto).