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Geodesics and Bounded Harmonic Functions on Infinite Graphs

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Random walk
 planar graphs
 geodesic rays
 harmonic functions
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1991
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SUNY Plattsburgh Mathematics Faculty Work
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http://hdl.handle.net/20.500.12648/1119
Abstract
It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitely-intersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph.
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Northshield, S. (1991). Geodesics and Bounded Harmonic Functions on Infinite Graphs. Proceedings of the American Mathematical Society, 113(1). http://doi.org/10.1090/S0002-9939-1991-1076576-5
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This article has been published in the September 1991 issue of Proceedings of the American Mathematical Society.
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