Exploration processes and SLE$_6$

TL;DR

This paper establishes the reversibility and scaling limits of radial exploration processes in hexagonal domains, showing they converge to radial SLE$_6$ and revealing new properties of SLE$_6$ traces.

Contribution

It proves the reversibility of radial exploration processes and their convergence to radial SLE$_6$, also linking full-plane SLE$_6$ as a scaling limit.

Findings

Reversibility of exploration processes from boundary to interior points.

Scaling limit of exploration processes is radial SLE$_6$.

Time-reversal of radial SLE$_6$ relates to full-plane 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.