Mathematische Annalen

, Volume 287, Issue 1, pp 613–626 | Cite as

Recurrence and transience of the edge graph of a tiling of the euclidean plane

  • P. M. Soardi
Article

Keywords

Euclidean Plane Edge Graph 
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References

  1. [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 1988Google Scholar
  2. [B] Breen, M.: Tilings whose members have finitely many neighbors. Isr. J. Math.52, 140–146 (1985)Google Scholar
  3. [C-W] Cartwright, D.I., Woess, W.: Infinite graphs with nonconstant Dirichlet finite harmonic functions. PreprintGoogle Scholar
  4. [D] Dodziuk, J.: Difference equations, isoperimetric inequalities and transience of certain random walks. Trans. Am. Math. Soc.284, 787–794 (1984)Google Scholar
  5. [Do] Doyle, P.: Electric currents in infinite networks. PreprintGoogle Scholar
  6. [G] Gerl, P.: Random walks on graphs with a strong isoperimetric inequality. J. Theor. Probab.1, 171–187 (1988)Google Scholar
  7. [G-S] Grünbaum, B., Shephard, G.C.: Tilings and patterns. New York: Freeman 1987Google Scholar
  8. [K-Y] Kayano, T., Yamasaki, M.: Boundary limits of discrete Dirichlet potentials. Hiroshima Math. J.14, 401–406 (1984)Google Scholar
  9. [K-S-K] Kemeny, G.K., Snell, J.L., Knapp, A.W.: Denumerable Markov chains. Berlin Heidelberg New York: Springer 1976Google Scholar
  10. [L] Lyons, T.: A simple criterion for transience of a reversible Markov chain. Ann. Probab.11, 393–402 (1984)Google Scholar
  11. [M] Mohar, B.: Isoperimetric numbers and spectral radius of some intinite planar graphs. PreprintGoogle Scholar
  12. [N-W] Nash-Willians, C. St. J.A.: Random walks and electrical currents in networks. Proc. Cam. Phil. Soc.55, 181–194 (1959)Google Scholar
  13. [McG] Mc Guinnes, S.: Recurrent networks and a theorem of Nash-Williams. PreprintGoogle Scholar
  14. [P] Polya, G.: Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann.84, 149–160 (1921)Google Scholar
  15. [S] Schlesinger, E.: Infinite networks and Markov chains. PreprintGoogle Scholar
  16. [S-W] Soardi, P.M., Woess, W.: Uniqueness of currents in infinite resistive networks. Discrete Appl. Math.Google Scholar
  17. [T] Thomassen, C.: Transient random walks, harmonic functions and electric currents in infinite resistive networks. PreprintGoogle Scholar
  18. [V] Varopoulos, N.Th.: Isoperimetric inequalities and Markov chains. J. Funct. Anal.63, 215–239 (1985)Google Scholar
  19. [Y1] Yamasaki, M.: Parabolic and hyperbolic infinite networks., Hiroshima Math. J.7, 135–146 (1977)Google Scholar
  20. [Y2] Yamasaki M.: Potentials on an infinite network. Mem. Fac. Shimane Univ.13, 31–44 (1979)Google Scholar
  21. [Z] Zemanian, A.H.: Infinite electrical networks. Proc. IEEE64, 6–17 (1976)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • P. M. Soardi
    • 1
  1. 1.Dipartimento di MatematicaUniversità di MilanoMilanoItaly

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