Transience of percolation clusters on wedges

Noam Berger (University of California, Los Angeles)
Itai Benjamini (The Weizmann Institute)
Omer Angel (University of British Columbia)
Yuval Peres (The University of California, Berkeley)

Abstract


We study random walks on supercritical percolation clusters on wedges in $Z^3$, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on $Z^2$ is equivalent to the (a.s.) existence of a finite energy flow on the supercritical percolation cluster. This answers a question of C. Hoffman.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 655-669

Publication Date: August 7, 2006

DOI: 10.1214/EJP.v11-345

References

  1. Antal, Peter; Pisztora, Agoston. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996), no. 2, 1036--1048. MR1404543 (98b:60168)
  2. Babson, Eric; Benjamini, Itai. Cut sets and normed cohomology with applications to percolation. Proc. Amer. Math. Soc. 127 (1999), no. 2, 589--597. MR1622785 (99g:05119)
  3. Benjamini, Itai; Pemantle, Robin; Peres, Yuval. Unpredictable paths and percolation. Ann. Probab. 26 (1998), no. 3, 1198--1211. MR1634419 (99g:60183)
  4. \bibitem{BeSc} Benjamini, I. and Schramm, O. (1998), Oriented random walk on the Heisenberg group and percolation. {\sl Unpublished manuscript}.
  5. Dembo, Amir; Gandolfi, Alberto; Kesten, Harry. Greedy lattice animals: negative values and unconstrained maxima. Ann. Probab. 29 (2001), no. 1, 205--241. MR1825148 (2002f:60087)
  6. Doyle, Peter G.; Snell, J. Laurie. Random walks and electric networks. Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, 1984. xiv+159 pp. ISBN: 0-88385-024-9 MR0920811 (89a:94023)
  7. Fortuin, C. M.; Kasteleyn, P. W.; Ginibre, J. Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 (1971), 89--103. MR0309498 (46 #8607)
  8. Grimmett, G. R.; Kesten, H.; Zhang, Y.. Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 (1993), no. 1, 33--44. MR1222363 (94i:60078)
  9. Gromov, Mikhael. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53, (1981), 53--73. MR0623534 (83b:53041)
  10. Häggström, Olle; Mossel, Elchanan. Nearest-neighbor walks with low predictability profile and percolation Ann. Probab. 26 (1998), no. 3, 1212--1231. MR1640343 (99h:60165)
  11. He, Zheng-Xu; Schramm, O. Hyperbolic and parabolic packings. Discrete Comput. Geom. 14 (1995), no. 2, 123--149. MR1331923 (96h:52017)
  12. Hoffman, Christopher. Energy of flows on $Z\sp 2$ percolation clusters. Random Structures Algorithms 16 (2000), no. 2, 143--155. MR1742348 (2001a:60115)
  13. Heicklen, Deborah; Hoffman, Christopher. Return probabilities of a simple random walk on percolation Electron. J. Probab. 10 (2005), no. 8, 250--302 (electronic). MR2120245 (2005j:60182)
  14. Hoffman, Christopher; Mossel, Elchanan. Energy of flows on percolation clusters. Potential Anal. 14 (2001), no. 4, 375--385. MR1825692 (2003b:60123)
  15. \bibitem{LP} Levin D. and Peres Y. (1998). Energy and cutsets in infinite percolation clusters. {\it Proceedings of the Cortona Workshop on Random Walks and Discrete Potential Theory}, M. Picardello and W. Woess (Editors), Cambridge Univ.\ Press.
  16. Liggett, T. M.; Schonmann, R. H.; Stacey, A. M. Domination by product measures. Ann. Probab. 25 (1997), no. 1, 71--95. MR1428500 (98f:60095)
  17. Lyons, Terry. A simple criterion for transience of a reversible Markov chain. Ann. Probab. 11 (1983), no. 2, 393--402. MR0690136 (84e:60102)
  18. Lyons, Russell; Pemantle, Robin; Peres, Yuval. Resistance bounds for first-passage percolation and maximum flow. J. Combin. Theory Ser. A 86 (1999), no. 1, 158--168. MR1682969 (2000c:60161)
  19. \bibitem{yvkl} Lyons R., with Peres Y., (2001) {\it Probability on trees and networks.} Cambridge University Press, in preparation. Current version available at %\hfill\break {\tt http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html}.
  20. Peres, Yuval. Probability on trees: an introductory climb. 193--280, Lecture Notes in Math., 1717, Springer, Berlin, 1999. MR1746302 (2001c:60139)
  21. Pólya, Georg. Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die (German) Math. Ann. 84 (1921), no. 1-2, 149--160. MR1512028
  22. Varopoulos, N. Th.; Saloff-Coste, L.; Coulhon, T. Analysis and geometry on groups. Cambridge Tracts in Mathematics, 100. Cambridge University Press, Cambridge, 1992. xii+156 pp. ISBN: 0-521-35382-3 MR1218884 (95f:43008)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.