Exploration Processes and SLE_{6}

Jianping Jiang

2017, v.23, №3, 445-466

ABSTRACT

We define radial exploration processes from $a$ to $b$ and from $b$ to $a$ in a domain $D$ of hexagons where $a$ is a boundary point and $b$ is an interior point. We prove the reversibility: the time-reversal of the process from $b$ to $a$ has the same distribution as the process from $a$ to $b$. We show the scaling limit of such an exploration process is a radial SLE$_6$ in $D$. As a consequence, the distribution of the last hitting point with the boundary of any radial SLE$_6$ is harmonic measure. We also prove the scaling limit of a similar exploration process defined in the full complex plane $\mathbb{C}$ is a full-plane SLE$_6$. A by-product of these results is that the time-reversal of a radial SLE$_6$ trace after the last visit to the boundary is a full-plane SLE$_6$ trace up to the first visit of the boundary.

%We define radial exploration processes in a domain of hexagons between a boundary point and an interior point. We prove the reversibility of such processes. We show that the scaling limit of such an exploration process is a radial SLE$_6$. We also prove that the scaling limit of a similar exploration process defined in the full complex plane is the full-plane SLE$_6$. As a consequence, the time-reversal of a radial SLE$_6$ trace after the last visit to the boundary is a full-plane SLE$_6$ trace up to the first visit of the boundary.

Keywords: exploration process, Schramm\tire Loewner evolution, scaling limit, harmonic measure, reversibility

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