Percolation on Penrose Tilings

Hof, A.

Canadian mathematical bulletin, Tome 41 (1998) no. 2, pp. 166-177

Résumé

In Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely 0 or 1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of ${{\mathbb{Z}}^{d}}$ , and to other percolation processes, including Bernoulli bond percolation.

Détail

DOI : 10.4153/CMB-1998-026-0

Mots-clés : 60K35, 82B43

Bibliographie

[1] 1. van den Berg, J. and Keane, M., On the continuity of the percolation function. In: Conference on Modern Analysis and Probability (Eds. R. Beals et al.). Contemp. Math. 26 (1984), 61–65. Google Scholar

[2] 2. de Bruijn, N. G., Algebraic theory of Penrose's non-periodic tilings of the plane. I. Nederl. Akad. Wetensch. Indag. Math. 84 (1981), 39–52. Google Scholar

[3] 3. de Bruijn, N. G., Algebraic theory of Penrose's non-periodic tilings of the plane. II. Nederl. Akad. Wetensch. Indag. Math. 84 (1981), 53–66. Google Scholar

[4] 4. de Bruijn, N. G., Updown generation of Penrose tilings. Indag. Math. (N.S.) 1 (1990), 201–220. Google Scholar

[5] 5. Burton, R. M. and Keane, M., Density and uniqueness in percolation. Commun. Math. Phys. 121 (1989), 501–505. Google Scholar

[6] 6. Fortuin, C. M., Ginibre, J., and Kasteleyn, P. W., Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22 (1971), 89–103. Google Scholar

[7] 7. Gandolfi, A., Keane, M. S., and Newman, C. M., Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92 (1992), 511–527. Google Scholar

[8] 8. Geerse, C. P. M. and Hof, A., Lattice gas models on self-similar aperiodic tilings. Rev. Math. Phys. 3 (1991), 163–221. Google Scholar

[9] 9. Grimmett, G., Percolation. Springer, 1989. Google Scholar

[10] 10. Gr ünbaum, B. and Shephard, G. C., Tilings and Patterns. W. H. Freeman and Company, 1987. Google Scholar

[11] 11. Hof, A., Some remarks on discrete aperiodic Schrödinger operators. J. Stat. Phys. 72 (1993), 1353–1374. Google Scholar

[12] 12. Hof, A., On diffraction by aperiodic structures. Commun. Math. Phys. 169 (1995), 25–43. Google Scholar

[13] 13. Lu, Jian Ping and Birman, J. L., Percolation and scaling on a quasilattice. J. Stat. Phys. 46 (1987), 1057–1066. Google Scholar

[14] 14. Katz, A. and Duneau, M., Quasiperiodic patterns and icosahedral symmetry. J. Physique 47 (1986), 181–196. Google Scholar

[15] 15. Kesten, H., Percolation Theory for Mathematicians. Birkhäuser, 1982. Google Scholar

[16] 16. Lunnon, W. F. and Pleasants, P. A. B., Quasicrystallographic tilings. J. Math. Pures Appl. 66 (1987), 217–263. Google Scholar

[17] 17. Menshikov, M. V., Coincidence of critical points in percolation problems. Soviet Math. Dokl. 33 (1986), 856–869. Google Scholar

[18] 18. Newman, C. M. and Schulman, L. S., Infinite clusters in percolation models. J. Stat. Phys. 26 (1981), 613–628. Google Scholar

[19] 19. Preston, C. J., Gibbs States on Countable Sets. Cambridge University Press, 1974. Google Scholar

[20] 20. Radin, C. and Wolf, M., Space tilings and local isomorphism. Geom. Dedicata 42 (1992), 355–360. Google Scholar

[21] 21. Robinson, E. A. Jr., The dynamical properties of Penrose tilings. Trans. Amer. Math. Soc. 348 (1996), 4447–4469. Google Scholar

[22] 22. Rudolph, D. J., Rectangular tilings of Rn and free Rn actions. Lecture Notes in Math. 1342 (1988), 653–689. Google Scholar

[23] 23. Sakamoto, S., Yonezawa, F., Aoki, K., Nosé, S., and Hori, M., Percolation in two-dimensional quasicrystals by Monte Carlo simulations. J. Phys. A 22(1989), L705–L709. Google Scholar

[24] 24. Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W., Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984), 1951–1953. Google Scholar

[25] 25. Solomyak, B., Dynamics of self-similar tilings. Ergodic Theory Dynamical Systems, to appear. Google Scholar

[26] 26. Walters, P., An Introduction to Ergodic Theory. Springer Verlag, 1982. Google Scholar

[27] 27. Yonezawa, F. et al., Percolation in Penrose tiling and its dual—in comparison with analysis for Kagomé, dice and square lattices. J. Non-Cryst. Solids 106 (1988), 262–269. Google Scholar