Abstract
Benjamini and Schramm (Invent Math 126(3):565–587, 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.
Highlights
A circle packing is a collection P of discs in the Riemann sphere C ∪ {∞} such that distinct discs in P do not overlap, but may be tangent
Given a circle packing P, its tangency graph is the graph whose vertices are the discs in P and where two vertices are connected by an edge if and only if their corresponding discs are tangent
The Circle Packing Theorem [24,39] states that every finite, simple1 planar graph may be represented as the tangency graph of a circle packing, and that if the graph is a triangulation the circle packing is unique up to Möbius transformations and reflections
Summary
A circle packing is a collection P of discs in the Riemann sphere C ∪ {∞} such that distinct discs in P do not overlap (i.e., have disjoint interiors), but may be tangent. [6,19,31,32,38] and references therein He and Schramm pioneered the use of circle packing to study probabilistic questions about planar graphs, showing in particular that a bounded degree, connected, planar triangulation is CP parabolic if and only if it is recurrent for simple random walk [20]. Angel et al [4] showed that every bounded harmonic function and every positive harmonic function on a bounded degree, connected, simple planar triangulation can be represented geometrically in terms of the triangulation’s circle packing in the unit disc. The difference in strength between these theorems is unsurprising given that the existence of non-constant harmonic Dirichlet functions is known to be stable under various perturbations of the underlying space [13,21,36], while the existence of non-constant bounded harmonic functions is known to be unstable in general under similar perturbations [7]
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