Harmonic Dirichlet Functions on Planar Graphs
Probability
2019-01-11 v2 Metric Geometry
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 and every circle packing of in a domain , there is a canonical, explicit bounded linear isomorphism between the space of harmonic Dirichlet functions on and the space of harmonic Dirichlet functions on .
Cite
@article{arxiv.1707.07751,
title = {Harmonic Dirichlet Functions on Planar Graphs},
author = {Tom Hutchcroft},
journal= {arXiv preprint arXiv:1707.07751},
year = {2019}
}
Comments
V2: Revised in accordance with referees' comments: errors eradicated, exposition expanded, corollaries contributed. Accepted version. To appear in Discrete and Computational Geometry