Harmonic Dirichlet Functions on Planar Graphs

Tom Hutchcroft

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arXiv · 1707.07751

Harmonic Dirichlet Functions on Planar Graphs

Tom Hutchcroft

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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 $D$.

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Tom Hutchcroft. 2017-07-24. Harmonic Dirichlet Functions on Planar Graphs. https://arxiv.org/abs/1707.07751

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