多弦 SLE$_{0+}$ 的大偏差、实有理函数与 Laplace 算子的 zeta 正则化行列式
Large deviations of multichordal SLE$_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians
AI总结：
本文证明了多弦 SLE$_{0+}$ 曲线的强极大偏差原理，引入了与 Laplace 算子 zeta 正则化行列式相关的 Loewner 势，并证明其最小化多弦为有理函数的实轨迹，同时作为副产品解析地证明了 Shapiro 猜想。
摘要：
我们证明了关于 Hausdorff 度量的多条弦 SLE$_{0+}$ 曲线的强极大偏差原理（LDP）。在单弦情形下，该结果加强了第二作者早先的部分结果。我们还引入了 Loewner 势，在光滑情形下，它具有用 Laplace 算子的 zeta 正则化行列式表示的简单表达式。该势与 LDP 速率函数仅相差一个仅依赖于边界数据的加性常数，该常数满足的偏微分方程是共形场论中中心荷 $c \to -\infty$ 时二阶 Belavin-Polyakov-Zamolodchikov 方程的半经典极限。此外，我们证明了在上半平面中对于给定边界数据使该势最小化的每条多弦都是有理函数的实轨迹且是唯一的，因此与多重 SLE$_κ$ 的 $κ\to 0+$ 极限相一致。作为副产品，我们给出了实枚举几何中 Shapiro 猜想的解析证明，该猜想最初由 Eremenko 和 Gabrielov 证明：若有理函数的所有临界点均为实数，则该函数在 Möbius 变换的复合下为实函数。
英文摘要：
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 $κ\to 0+$ limit of the multiple SLE$_κ$. 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 Möbius transformation.
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