Exploration processes and SLE$_6$

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.