Discrete Harmonic Functions in the Three-Quarter Plane

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.

Quantitative recurrence results for random walks

First, we prove an almost sure local central limit theorem for lattice random walks in the plane. The corresponding version for random walks in the line has been considered previously by the author. This gives us an extension of Pólya’s Recurrence Theorem, namely we consider an appropriate subsequence of the random walk and give the asymptotic number of returns to the origin and other states. Secondly, we prove an almost sure local central limit theorem for (not necessarily lattice) random walks in the line or in the plane, which will also give us quantitative recurrence results. Finally, we prove a version of the almost sure central limit theorem for multidimensional random walks. This is done by exploiting a technique developed by the author.