## Abstract

We prove a strong large deviation principle (LDP) for multiple chordal SLE0+SLE0+ 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, which satisfies PDEs arising as a semiclassical limit of the Belavin–Polyakov–Zamolodchikov equations of level 2 in conformal field theory with central charge c→−∞c→−∞.

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 κ→0+κ→0+ limit of the multiple SLEκSLEκ. 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 with a Möbius transformation.

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 κ→0+κ→0+ limit of the multiple SLEκSLEκ. 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 with a Möbius transformation.

Original language | English |
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Journal | JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY |

Early online date | 28 Apr 2023 |

DOIs | |

Publication status | E-pub ahead of print - 28 Apr 2023 |

MoE publication type | A1 Journal article-refereed |