Rates of Convergence for the Planar Discrete Green's Function in Pacman Domains

Abstract

We obtain upper bounds for the rates of convergence for the simple random walk Green's function in the domains $D_\alpha = D_{\alpha}(n)=\{re^{i\theta}\in \mathbb{C}:0 <\theta<2\pi-\alpha, 0<r<2n\}-z_0,$ where $z_0\in\mathbb{Z}^2$ is a point closest to $ne^{i(\pi-\alpha/2)}$. The rate depends on the angle of the wedge and is what was suggested by the sharpest available results in the extreme cases $\alpha =0$ and $\alpha=\pi$. Our proof uses the KMT coupling between random walk and Brownian motion.