#### Document Type

Conference Paper

#### Publication Date

2022

#### Publication Source

Proceedings of the 2021 Undergraduate Mathematics Day

#### Volume

7

#### Abstract

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.

#### Keywords

Circle packing, Conformal invariance, Harmonic measure distribution function

#### Disciplines

Mathematics

#### eCommons Citation

Wilson, Ella, "Using Circle Packings to Approximate Harmonic Measure Distribution Functions" (2022). *Undergraduate Mathematics Day: Proceedings and Other Materials*. 46.

https://ecommons.udayton.edu/mth_epumd/46

*Abstract only*

## Comments

Presented at University of Dayton Undergraduate Mathematics Day Nov. 6, 2021.