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Seminar
FIELD Math:Analysis
DATE August 03 (Tue), 2021
TIME 10:00-11:00
PLACE Online
SPEAKER Zhan, Dapeng
HOST Kang, Nam-Gyu
INSTITUTE Michigan State University
TITLE Two-curve Green’s function for Schramm-Loewner evolution
ABSTRACT

Schramm-Loewner evolution (SLE$_\kappa$) is a one-parameter family of random fractal curves growing in plane domains. Given an SLE$_\kappa$ curve $\gamma$ with $\kappa\in(0,8)$ in a domain $D$ and a point $z_0$, the Green's function for $\gamma$ at $z_0$ is the limit $G(z_0):= \lim_{r\downarrow 0} r^{-\alpha} \mathbb{P}[\mbox{dist}(z_0,\gamma) \less r]$, for some positive exponent $\alpha$, if the limit converges and is not trivial, i.e., not zero. There are several variations. The $z_0$ may be an interior point or a boundary point of $D$. In the former case $\alpha$ is related to the Hausdorff dimension $d$ of $\gamma$ by $\alpha=2-d=1-\frac\kappa 8$. The single point $z_0$ may be replaced by a number of points $z_1,\dots, z_n$.

In this talk, I will describe a method of proving the existence of Green's function for a pair of SLE$_\kappa$ curves $(\gamma_1,\gamma_2)$ in a multiple-SLE$_\kappa$ configuration. The function at a point $z_0$ is the limit $ \lim_{r\downarrow 0} r^{-\alpha} \mathbb{P}[\mbox{dist}(z_0,\gamma_j)\less r, j=1,2]$. The $z_0$ could be an interior point or a boundary point. The method can also be used to derive a Green's function related to cut-points of a single SLE curve.

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