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