# Discrete harmonic functions in the three-quarter plane

## TL;DR

This paper extends the boundary value problem approach to find explicit formulas for positive discrete harmonic functions in the three-quarter plane, a domain less studied than the quarter plane, using conformal mappings.

## Contribution

It introduces a novel method for solving harmonic functions in the three-quarter plane by splitting the domain into two convex cones, generalizing previous quarter plane techniques.

## Key findings

- Derived explicit algebraic generating functions for harmonic functions
- Extended boundary value problem methods to three-quarter plane
- Provided new insights into random walks avoiding a quadrant

## Abstract

In this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane -- resolution of a functional equation via boundary value problem using a conformal mapping -- to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.