Using Circle Packings to Approximate Harmonic Measure Distribution Functions

Harmonic measure distribution functions, h-functions, encode information about the geometry of domains in the plane. Specifically, given a domain and a basepoint in the domain, for a fixed radius, r, the value h(r) is the probability that a Brownian particle first exits the domain within distance r of the basepoint. There are many domains for which we can compute h-functions, such as the disk and the inside and outside of a wedge. However, exact computation is often difficult or impossible for more complicated domains, so we need methods to approximate these h-functions. In this paper, we develop two methods for approximating h-functions using circle packings to discretize domains. We also discuss connections to open questions in the field of h-functions