Phase Transition for a Family of Complex-driven Loewner Hulls

Abstract

Building on H. Tran's study of Loewner hulls generated by complex-valued driving functions, which showed the existence of a phase transition, we answer the question of whether the phase transition for complex-driven hulls matches the phase transition for real-driven hulls. This is accomplished through a detailed study of the Loewner hulls generated by driving functions $c\sqrt{1-t}$ and $c\sqrt{\tau +t}$ for $c\in \mathbb{C}$ and $\tau \ge 0$. This family also provides examples of new geometric behavior that is possible for complex-driven hulls but prohibited for real-driven hulls.

References in corpus (3)

• A Sharp Condition for the Loewner Equation to Generate Slits
• Exact Solutions for Loewner Evolutions
• Loewner Equation driven by complex-valued functions