Trace for the Loewner Equation with Singular Forcing

Abstract

The Loewner equation describes the time development of an analytic map into the upper half of the complex plane in the presence of a "forcing", a defined singularity moving around the real axis. The applications of this equation use the trace, the locus of singularities in the upper half plane. This note discusses the structure of the trace for the case in which the forcing function, xi(t), is proportional to (-t)^beta with beta in the interval (0, 1/2). In this case, the trace is a simple curve, gamma(t), which touches the real axis twice. It is computed by using matched asymptotic analysis to compute the trajectory of the Loewner evolution in the neighborhood of the singularity, and then assuming a smooth mapping of these trajectories away from the singularity. Near the t=0 singularity, the trace has a shape given by [ Re(gamma(t)-gamma(0)) ]^(1-beta) ~ [ beta*Im(gamma(t)) ]^beta ~ O(xi(t))^(1-beta). A numerical calculation of the trace provides support for the asymptotic theory.