Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Geodesics and bounded harmonic functions on infinite planar graphs


Author: S. Northshield
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] A. Ancona, Positive harmonic functions and hyperbolicity, Potential Theory (Kràl et al.) Lecture Notes in Math., vol. 1344, Springer-Verlag, Berlin, 1988, pp. 1-23. MR 973878
  • [2] P. Cartier, Fonctions harmoniques sur un Arbre, Symposia Mathematica, vol. 9, Academic Press, New York, 1982, pp. 203-270. MR 0353467 (50:5950)
  • [3] W. S. Kendall, Brownian motion on a surface of negative curvature, Séminaire de Probabililités 18, 1982/1983 (J. Azéma and M. Yor, eds.), Lecture Notes in Math., vol. 1059, Springer-Verlag, Berlin, 1984, pp. 70-76. MR 770949 (86f:58170)
  • [4] S. Northshield, Schrödinger operators on infinite graphs, doctoral dissertation, Univ. of Rochester, 1989.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60J15

Retrieve articles in all journals with MSC: 60J15


Additional Information

DOI: https://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

American Mathematical Society