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

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 Möbius transformation.

arXiv comment67 pages, 4 figures. Final version to appear in JEMS

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Cited by in corpus (4)

• The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles
• From the Conformal Anomaly to the Virasoro Algebra
• Two-loop Loewner potentials
• Minimal degree rational open up mappings and related questions