Phase Transition for a Family of Complex-driven Loewner Hulls

TL;DR

This paper investigates phase transitions in complex-driven Loewner hulls, comparing them to real-driven cases, and explores new geometric behaviors unique to complex driving functions.

Contribution

It demonstrates that phase transitions in complex-driven hulls can differ from real-driven ones and introduces new geometric phenomena specific to complex driving functions.

Findings

Phase transition behavior differs between complex and real-driven hulls.

Identification of new geometric behaviors in complex-driven Loewner hulls.

Analysis of specific complex driving functions revealing unique hull properties.

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 $\tau \geq 0$. This family also provides examples of new geometric behavior that is possible for complex-driven hulls but prohibited for real-driven hulls.