Potential Theory and the Boundary of Combinatorial Graphs

TL;DR

This paper develops a potential theory framework for finite graphs, establishing discrete analogues of classical continuous results like hitting times, inequalities, and measure support, thus bridging graph theory and potential theory.

Contribution

It introduces a well-behaved notion of boundary for graphs and proves several classical potential theory results in this discrete setting, expanding the theoretical understanding of graph boundaries.

Findings

Random walk hits boundary within quadratic time of diameter

Boundary measures maximize certain integral functionals

Discrete inequalities analogous to classical potential theory results

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.