References
[A] Ancona, A.: Positive harmonic functions and hyperbolicity. In: Potential theory, survey and problems. Lect. Notes Math., vol. 1344, pp 1–23. Berlin Heidelberg New York: Springer 1988
[B] Breen, M.: Tilings whose members have finitely many neighbors. Isr. J. Math.52, 140–146 (1985)
[C-W] Cartwright, D.I., Woess, W.: Infinite graphs with nonconstant Dirichlet finite harmonic functions. Preprint
[D] Dodziuk, J.: Difference equations, isoperimetric inequalities and transience of certain random walks. Trans. Am. Math. Soc.284, 787–794 (1984)
[Do] Doyle, P.: Electric currents in infinite networks. Preprint
[G] Gerl, P.: Random walks on graphs with a strong isoperimetric inequality. J. Theor. Probab.1, 171–187 (1988)
[G-S] Grünbaum, B., Shephard, G.C.: Tilings and patterns. New York: Freeman 1987
[K-Y] Kayano, T., Yamasaki, M.: Boundary limits of discrete Dirichlet potentials. Hiroshima Math. J.14, 401–406 (1984)
[K-S-K] Kemeny, G.K., Snell, J.L., Knapp, A.W.: Denumerable Markov chains. Berlin Heidelberg New York: Springer 1976
[L] Lyons, T.: A simple criterion for transience of a reversible Markov chain. Ann. Probab.11, 393–402 (1984)
[M] Mohar, B.: Isoperimetric numbers and spectral radius of some intinite planar graphs. Preprint
[N-W] Nash-Willians, C. St. J.A.: Random walks and electrical currents in networks. Proc. Cam. Phil. Soc.55, 181–194 (1959)
[McG] Mc Guinnes, S.: Recurrent networks and a theorem of Nash-Williams. Preprint
[P] Polya, G.: Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann.84, 149–160 (1921)
[S] Schlesinger, E.: Infinite networks and Markov chains. Preprint
[S-W] Soardi, P.M., Woess, W.: Uniqueness of currents in infinite resistive networks. Discrete Appl. Math.
[T] Thomassen, C.: Transient random walks, harmonic functions and electric currents in infinite resistive networks. Preprint
[V] Varopoulos, N.Th.: Isoperimetric inequalities and Markov chains. J. Funct. Anal.63, 215–239 (1985)
[Y1] Yamasaki, M.: Parabolic and hyperbolic infinite networks., Hiroshima Math. J.7, 135–146 (1977)
[Y2] Yamasaki M.: Potentials on an infinite network. Mem. Fac. Shimane Univ.13, 31–44 (1979)
[Z] Zemanian, A.H.: Infinite electrical networks. Proc. IEEE64, 6–17 (1976)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Soardi, P.M. Recurrence and transience of the edge graph of a tiling of the euclidean plane. Math. Ann. 287, 613–626 (1990). https://doi.org/10.1007/BF01446917
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01446917