Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 15(2010) > Paper 45 open journal systems 


Plaquettes, Spheres, and Entanglement

Geoffrey R Grimmett, University of Cambridge
Alexander E Holroyd, Microsoft Research


Abstract
The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d-1)-sphere and encloses the origin. This is proved by a path-counting argument in a dual model. When d=3, this permits an improved lower bound on the critical point pe of entanglement percolation, namely pe ≥ μ-2 where μ is the connective constant for self-avoiding walks on Z3. Furthermore, when the edge density p is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail.


Full text: PDF

Pages: 1415-1428

Published on: September 19, 2010
Bibliography
  1. Aizenman, Michael; Barsky, David J. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (1987), no. 3, 489--526. MR0874906 (88c:82026)
  2. Aizenman, M.; Chayes, J. T.; Chayes, L.; Fröhlich, J.; Russo, L. On a sharp transition from area law to perimeter law in a system of random surfaces. Comm. Math. Phys. 92 (1983), no. 1, 19--69. MR0728447 (85d:82006)
  3. Aizenman, Michael; Grimmett, Geoffrey. Strict monotonicity for critical points in percolation and ferromagnetic models. J. Statist. Phys. 63 (1991), no. 5-6, 817--835. MR1116036 (92i:82060)
  4. Atapour, Mahshid; Madras, Neal. On the number of entangled clusters. J. Stat. Phys. 139 (2010), no. 1, 1--26. MR2602981
  5. Barsky, David J.; Grimmett, Geoffrey R.; Newman, Charles M. Percolation in half-spaces: equality of critical densities and continuity of the percolation probability. Probab. Theory Related Fields 90 (1991), no. 1, 111--148. MR1124831 (92m:60086)
  6. Bezuidenhout, Carol; Grimmett, Geoffrey. The critical contact process dies out. Ann. Probab. 18 (1990), no. 4, 1462--1482. MR1071804 (91k:60111)
  7. Dirr, N.; Dondl, P. W.; Grimmett, G. R.; Holroyd, A. E.; Scheutzow, M. Lipschitz percolation. Electron. Commun. Probab. 15 (2010), 14--21. MR2581044
  8. Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6 MR1707339 (2001a:60114)
  9. Grimmett, Geoffrey; Hiemer, Philipp. Directed percolation and random walk. In and out of equilibrium (Mambucaba, 2000), 273--297, Progr. Probab., 51, Birkhäuser Boston, Boston, MA, 2002. MR1901958 (2003b:60159)
  10. Grimmett, Geoffrey R.; Holroyd, Alexander E. Entanglement in percolation. Proc. London Math. Soc. (3) 81 (2000), no. 2, 485--512. MR1770617 (2001j:60185)
  11. Grimmett, Geoffrey R.; Holroyd, Alexander E. Geometry of Lipschitz percolation. Preprint, arXiv:1007.3762 (2010).
  12. Grimmett, G. R.; Marstrand, J. M. The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 (1990), no. 1879, 439--457. MR1068308 (91m:60186)
  13. Häggström, Olle. Uniqueness of the infinite entangled component in three-dimensional bond percolation. Ann. Probab. 29 (2001), no. 1, 127--136. MR1825145 (2002a:60159)
  14. Holroyd, Alexander E. Existence of a phase transition for entanglement percolation. Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 2, 231--251. MR1765912 (2001k:82051)
  15. Holroyd, Alexander E. Entanglement and rigidity in percolation models. In and out of equilibrium (Mambucaba, 2000), 299--307, Progr. Probab., 51, Birkhäuser Boston, Boston, MA, 2002. MR1901959 (2003b:60161)
  16. Holroyd, Alexander E. Inequalities in entanglement percolation. J. Statist. Phys. 109 (2002), no. 1-2, 317--323. MR1927926 (2003h:82044)
  17. Kantor, Yacov; Hassold, Gregory N. Topological entanglements in the percolation problem. Phys. Rev. Lett. 60 (1988), no. 15, 1457--1460. MR0935098 (89b:82001)
  18. Menʹshikov, M. V. Coincidence of critical points in percolation problems. (Russian) Dokl. Akad. Nauk SSSR 288 (1986), no. 6, 1308--1311. MR0852458 (88k:60175)
  19. Menʹshikov, M. V.; Molchanov, S. A.; Sidorenko, A. F. Percolation theory and some applications. (Russian) Translated in J. Soviet Math. 42 (1988), no. 4, 1766--1810. Itogi Nauki i Tekhniki, Probability theory. Mathematical statistics. Theoretical cybernetics, Vol. 24 (Russian), 53--110, i, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986. MR0865158 (88m:60273)
  20. Pönitz, André; Tittmann, Peter. Improved upper bounds for self-avoiding walks in $Zsp d$. Electron. J. Combin. 7 (2000), Research Paper 21, 10 pp. (electronic). MR1754251 (2000m:05015)
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489