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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Geodesics and bounded harmonic functions on infinite planar graphs
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by S. Northshield
Proc. Amer. Math. Soc. 113 (1991), 229-233
DOI: https://doi.org/10.1090/S0002-9939-1991-1076576-5

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.
References
  • Alano Ancona, Positive harmonic functions and hyperbolicity, Potential theory—surveys and problems (Prague, 1987) Lecture Notes in Math., vol. 1344, Springer, Berlin, 1988, pp. 1–23. MR 973878, DOI 10.1007/BFb0103341
  • P. Cartier, Fonctions harmoniques sur un arbre, Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità & Convegno di Teoria della Turbolenza, INDAM, Rome, 1971) Academic Press, London, 1972, pp. 203–270 (French). MR 0353467
  • Wilfrid S. Kendall, Brownian motion on a surface of negative curvature, Seminar on probability, XVIII, Lecture Notes in Math., vol. 1059, Springer, Berlin, 1984, pp. 70–76. MR 770949, DOI 10.1007/BFb0100032
  • S. Northshield, Schrödinger operators on infinite graphs, doctoral dissertation, Univ. of Rochester, 1989.
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Bibliographic Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 113 (1991), 229-233
  • MSC: Primary 60J15
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1076576-5
  • MathSciNet review: 1076576