Abstract.
We prove uniqueness of the infinite rigid component for standard bond percolation on periodic lattices in d-dimensional Euclidean space for arbitrary d, and more generally when the lattice is a quasi-transitive and amenable graph. Our approach to uniqueness of the infinite rigid component improves earlier ones, that were confined to planar settings.
References
Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111, 505–531 (1987)
Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Group-invariant percolation on graphs. Geom. Funct. Anal. 9, 29–66 (1999)
Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29, 1–65 (2001)
van den Berg, J., Keane, M.: On the continuity of the percolation function. In Conference in modern analysis and probability, R. Beals et al. (eds.), American Mathematical Society, Providence, RI, 1984, pp. 61–65
Burton, R., Keane, M.: Density and uniqueness in percolation. Comm. Math. Phys. 121, 501–505 (1989)
Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. Am. Math. Soc., Providence, RI, 1993
Grimmett, G.R.: The stochastic random-cluster process, and the uniqueness of random-cluster measures. Ann. Probab. 23, 1461–1510 (1995)
Grimmett, G.R.: Percolation, 2nd ed. Springer, New York, 1999
Häggström, O.: Uniqueness in two-dimensional rigidity percolation. Math. Proc. Cambridge Phil. Soc. 130, 175–188 (2001)
Häggström, O.: Uniqueness of the infinite entangled component in three-dimensional bond percolation. Ann. Probab. 29, 127–136 (2001)
Häggström, O., Peres, Y.: Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously. Probab. Th. Rel. Fields 113, 273–285 (1999)
Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Phil. Soc. 56, 13–20 (1960)
Holroyd, A.E.: Existence and uniqueness of infinite components in generic rigidity percolation. Ann. Appl. Probab. 8, 944–973 (1998)
Holroyd, A.E.: Percolation Beyond Connectivity. Ph.D. thesis, University of Cambridge, 1999
Holroyd, A.E.: Rigidity percolation and boundary conditions. Ann. Appl. Probab. 11, 1063–1078 (2001)
Jacobs, D.J., Thorpe, M.F.: Generic rigidity percolation in two dimensions. Phys. Rev. E 53, 3682–3693 (1996)
Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab. 25, 71–95 (1997)
Newman, C.M., Schulman, L.S.: Infinite clusters in percolation models. J. Statist. Phys. 26, 613–628 (1981)
Schonmann, R.H.: Stability of infinite clusters in supercritical percolation. Probab. Th. Rel. Fields 113, 287–300 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by the Swedish Research Council
Mathematics Subject Classification (2000): 60K35, 82B43
Rights and permissions
About this article
Cite this article
Häggström, O. Uniqueness of infinite rigid components in percolation models: the case of nonplanar lattices. Probab. Theory Relat. Fields 127, 513–534 (2003). https://doi.org/10.1007/s00440-003-0290-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-003-0290-2
Keywords
- Euclidean Space
- Periodic Lattice
- Percolation Model
- Planar Setting
- Bond Percolation