Harmonic Dirichlet Functions on Planar Graphs

TL;DR

This paper refines the understanding of harmonic Dirichlet functions on planar graphs by establishing a canonical isomorphism with functions on their circle packing domains, enhancing the connection between graph theory and complex analysis.

Contribution

It introduces an explicit, canonical isomorphism linking harmonic Dirichlet functions on planar graphs to those on their circle packing domains, extending prior results.

Findings

Established a canonical isomorphism between harmonic Dirichlet functions on graphs and domains

Extended Benjamini and Schramm's circle packing results to explicit function spaces

Provided a new tool for analyzing harmonic functions via geometric representations

Abstract

Benjamini and Schramm (1996) used circle packing to prove that every transient, bounded degree planar graph admits non-constant harmonic functions of finite Dirichlet energy. We refine their result, showing in particular that for every transient, bounded degree, simple planar triangulation $T$ and every circle packing of $T$ in a domain $D$, there is a canonical, explicit bounded linear isomorphism between the space of harmonic Dirichlet functions on $T$ and the space of harmonic Dirichlet functions on $D$.