On large deviations of SLEs, real rational functions, and zeta-regularized determinants of Laplacians

Abstract: The talk concerns a large deviation principle (LDP) for (multiple) Schramm-Loewner evolution (SLE) curves for the Hausdorff metric. When studying the LDP, we introduced a ''Loewner potential'' that describes the rate function. This object turned out to have several intrinsic, and perhaps surprising, connections to various fields. For instance, it has a simple expression in terms of zeta-regularized determinants of Laplace-Beltrami operators. On the other hand, minima of the Loewner potential solve a nonlinear first order PDE that arises in a semiclassical limit of certain correlation functions in conformal field theory (arguably also related to isomonodromic systems). Finally, the Loewner potential minimizers classify rational functions with real critical points, thereby providing a novel proof for a version of the now well-known Shapiro-Shapiro conjecture in real enumerative geometry. This talk is based on joint work with Yilin Wang (MIT).

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

Video recordings of talks are posted online at www.youtube.com/channel/UC0CXjG-ZSIZHy0S40Px2FEQ .