Potential Theory and the Boundary of Combinatorial Graphs

Abstract

Let $G=(V,E)$ be a finite, connected graph. We investigate a notion of boundary $\partial G \subseteq V$ and argue that it is well behaved from the point of view of potential theory. This is done by proving a number of discrete analogous of classical results for compact domains $\Omega \subset \mathbb{R}^d$. These include (1) an analogue of P\'olya's result that a random walk in $\Omega$ typically hits the boundary $\partial \Omega$ within $\lesssim \mbox{diam}(\Omega)^2$ units of time, (2) an analogue of the Faber-Krahn inequality, (3) an analogue of the Hardy inequality, (4) an analogue of the Alexandrov-Bakelman-Pucci estimate, (5) a stability estimate for hot spots and (6) a Theorem of Bj\"orck stating that probability measures $\mu$ that maximize $\int_{\Omega \times \Omega} \|x-y\|^{\alpha} d\mu(x) d\mu(y)$ are fully supported in the boundary.