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2021 Two-curve Green’s function for 2-SLE: the boundary case
Dapeng Zhan
Electron. J. Probab. 26: 1-58 (2021). DOI: 10.1214/21-EJP592

## Abstract

We prove that for $\upkappa \in (0,8)$, if $({\upeta _{1}},{\upeta _{2}})$ is a 2-SLE${_{\upkappa }}$pair in a simply connected domain D with an analytic boundary point ${z_{0}}$, then as $r\to {0^{+}}$, $\mathbb{P}[\mathrm{dist}({z_{0}},{\upeta _{j}})\textless r,j=1,2]$ converges to a positive number for some $\upalpha \textgreater 0$, which is called the two-curve Green’s function. The exponent α equals $\frac{12}{\upkappa }-1$ or $2(\frac{12}{\upkappa }-1)$ depending on whether ${z_{0}}$ is one of the endpoints of ${\upeta _{1}}$ or ${\upeta _{2}}$. We also find the convergence rate and the exact formula for the Green’s function up to a multiplicative constant. To derive these results, we construct two-dimensional diffusion processes and use orthogonal polynomials to obtain their transition density.

## Citation

Dapeng Zhan. "Two-curve Green’s function for 2-SLE: the boundary case." Electron. J. Probab. 26 1 - 58, 2021. https://doi.org/10.1214/21-EJP592

## Information

Received: 22 June 2020; Accepted: 30 January 2021; Published: 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1214/21-EJP592

Subjects:
Primary: 30C, 60G

JOURNAL ARTICLE
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Vol.26 • 2021
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