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.
Highlights
In the two-dimensional continuous case, a continuous harmonic function is a function for which the standard Laplacian2 is zero
We develop a systematic approach to find positive discrete harmonic functions in the three-quarter plane with Dirichlet conditions
We find an explicit expression for generating functions of discrete harmonic functions associated to random walks avoiding a quadrant with a mixed approach of [19] and [20]
Summary
In the two-dimensional continuous case, a continuous harmonic function is a function for which the standard Laplacian. In the cone exp of opening angle , sin is the unique (up to multiplicative constants) positive harmonic function equal to zero on the boundary. In the two-dimensional discrete case, consider the simplest Laplacian operator. The function is positive harmonic in the right half-plane and equal to zero on -axis.
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