The Loewner Equation for Multiple Hulls

TL;DR

This paper proves a conjecture that multiple hulls generated by the Loewner equation can be produced by a single oscillating function and extends the result to multiple continuous driving functions, providing a simulation method.

Contribution

It confirms the conjecture for two hulls and generalizes the result to multiple hulls with continuous driving functions, also offering a simulation approach.

Findings

Proved the conjecture for two hulls.

Generalized to multiple continuous driving functions.

Provided a simulation method for multiple hulls.

Abstract

Kager, Nienhuis, and Kadanoff conjectured that the hull generated from the Loewner equation driven by two constant functions with constant weights could be generated by a single rapidly and randomly oscillating function. We prove their conjecture and generalize to multiple continuous driving functions. In the process, we generalize to multiple hulls a result of Roth and Schleissinger that says multiple slits can be generated by constant weight functions. The proof gives a simulation method for hulls generated by the multiple Loewner equation.