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Phase transition for a family of complex-driven Loewner hulls

### Joan Lind and Jeffrey Utley

Vol. 15 (2022), No. 3, 447–474
##### 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 $c\in ℂ$ 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.

##### Keywords
Loewner evolution, Loewner hulls
Primary: 30C35