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Proceedings of the American Mathematical Society

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Geodesics and bounded harmonic functions on infinite planar graphs


Author: S. Northshield
Journal: Proc. Amer. Math. Soc. 113 (1991), 229-233
MSC: Primary 60J15
MathSciNet review: 1076576
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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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  • [4] S. Northshield, Schrödinger operators on infinite graphs, doctoral dissertation, Univ. of Rochester, 1989.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1076576-5
Keywords: Random walk, planar graphs, geodesic rays, harmonic functions
Article copyright: © Copyright 1991 American Mathematical Society