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.

中文翻译：

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四分之三平面中的离散谐波函数

在本文中，我们有兴趣在三个象限中找到具有Dirichlet条件的正离散谐波函数。尽管已经很好地研究了象限中的平面晶格（随机）游走，但是最近开发了避开象限的游走情况。我们将四分之一平面（通过使用保形映射通过边值问题解决泛函方程的解析）的方法扩展到四分之三平面，应用了将域划分为两个对称凸锥的策略。我们获得了与避免随机象限的随机游走相关的谐波函数的代数生成函数的简单显式表达式。