with Boundary Points, Multiple SLEs, and Log-Gases arXiv: - PowerPoint PPT Presentation

Gaussian Free Fields with Boundary Points, Multiple SLEs, and Log-Gases arXiv: math.PR/1903.09925v2 Makoto KATORI (Chuo Univ., Tokyo) joint work with Shinji KOSHIDA (Chuo Univ.) The 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI) Stochastic Analysis, Random Fields and Integrable Probability Kyushu University, Fukuoka 1 August 2019 1

Plan 1. Introduction 1.1 Stochastic log-gases in R 1.2 Loewner equation for multi-slit 1.3 Multiple Schramm-Leowner evolution (SLE) 1.4 Gaussian free field (GFF) 1.5 Imaginary surface 2. Imaginary Surface with Boundary Points (IS-BPs) 3. Two Ways of Sampling IS-BPs 4. Main Theorems (Theorems 4.1 and 4.2) 5. Proof of Theorem 4.1 6. Concluding Remarks 2

1. Introduction 1.1 Stochastic log-gases in R 3

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1.2 Loewner equation for multi-slit 7

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1.3 Multiple Schramm-Leowner evolution (multiple SLE) 10

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1.4 Gaussian free field (GFF) 13

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On the Green’s funciton 23

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1.5 Imaginary surface (IS) 25

2. Imaginary Surface with Boundary Points (IS-BPs) 26

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3. Two Ways of Sampling IS-BPs 31

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Coupling GFF and multiple SLE 34

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4. Main Theorems 37

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5. Proof of Theorem 4.1 42

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three conditions 44

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Proof of Theorem 4.1 47

Proof of Theorem 4.1 Dirichlet energy 48

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6. Concluding Remarks 50

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Thank you very much for your attention. 54