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 c1-t and cτ + t for c ∈ C and τ ≥ 0. This family also provides examples of new geometric behavior that is possible for complex-driven hulls but prohibited for real-driven hulls.
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