Large deviations of multichordal SLE$_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians

TL;DR

This paper establishes a large deviation principle for multichordal SLE curves, introduces a Loewner potential linked to zeta-regularized determinants, and connects minimal potential configurations to real rational functions, providing new insights in probability, geometry, and mathematical physics.

Contribution

It proves a strong large deviation principle for multichordal SLE curves, introduces a novel Loewner potential related to determinants, and links potential minimizers to real rational functions, also proving the Shapiro conjecture analytically.

Findings

Established a strong large deviation principle for multichordal SLE curves.

Introduced a Loewner potential expressed via zeta-regularized determinants.

Proved that potential-minimizing multichords are real rational functions, confirming the Shapiro conjecture.

Abstract

We prove a strong large deviation principle (LDP) for multiple chordal SLE$_{0+}$ curves with respect to the Hausdorff metric. In the single-chord case, this result strengthens an earlier partial result by the second author. We also introduce a Loewner potential, which in the smooth case has a simple expression in terms of zeta-regularized determinants of Laplacians. This potential differs from the LDP rate function by an additive constant depending only on the boundary data, that satisfies PDEs arising as a semiclassical limit of the Belavin-Polyakov-Zamolodchikov equations of level two in conformal field theory with central charge $c \to -\infty$. Furthermore, we show that every multichord minimizing the potential in the upper half-plane for given boundary data is the real locus of a rational function and is unique, thus coinciding with the $\kappa \to 0+$ limit of the multiple SLE$_\kappa$. As a by-product, we provide an analytic proof of the Shapiro conjecture in real enumerative geometry, first proved by Eremenko and Gabrielov: if all critical points of a rational function are real, then the function is real up to post-composition by a M\"obius transformation.