Large deviations of multichordal SLE$_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866913576847933440 |
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| author | Peltola, Eveliina Wang, Yilin |
| author_facet | Peltola, Eveliina Wang, Yilin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2006_08574 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Large deviations of multichordal SLE$_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians Peltola, Eveliina Wang, Yilin Mathematical Physics Complex Variables Probability 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. |
| title | Large deviations of multichordal SLE$_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians |
| topic | Mathematical Physics Complex Variables Probability |
| url | https://arxiv.org/abs/2006.08574 |