Electronic Communications in Probability

Percolation Beyond $Z^d$, Many Questions And a Few Answers

Itai Benjamini and Oded Schramm

Full-text: Open access


A comprehensive study of percolation in a more general context than the usual $Z^d$ setting is proposed, with particular focus on Cayley graphs, almost transitive graphs, and planar graphs. Results concerning uniqueness of infinite clusters and inequalities for the critical value $p_c$ are given, and a simple planar example exhibiting uniqueness and non-uniqueness for different $p>p_c$ is analyzed. Numerous varied conjectures and problems are proposed, with the hope of setting goals for future research in percolation theory.

Article information

Electron. Commun. Probab., Volume 1 (1996), paper no. 8, 71-82.

Accepted: 8 October 1996
First available in Project Euclid: 25 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B43: Percolation [See also 60K35]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Percolation criticality planar graph transitive graph isoperimetericinequality

This work is licensed under aCreative Commons Attribution 3.0 License.


Benjamini, Itai; Schramm, Oded. Percolation Beyond $Z^d$, Many Questions And a Few Answers. Electron. Commun. Probab. 1 (1996), paper no. 8, 71--82. doi:10.1214/ECP.v1-978. https://projecteuclid.org/euclid.ecp/1453756499

Export citation


  • M. Aizenman and G. R. Grimmett, (1991), Strict monotonicity for critical points in percolation and ferromagnetic models J. Statist. Phys. 63, 817–835.
  • E. Babson and I. Benjamini (1995), Cut sets in Cayley graphs, preprint.
  • I. Benjamini and Y. Peres (1994), Markov chains indexed by trees, Ann. Prob. 22, 219–243.
  • I. Benjamini and O. Schramm (1996), Conformal invariance of Voronoi percolation, preprint. Abstract and PostScript.
  • R. M. Burton and M. Keane (1989), Density and uniqueness in percolation, Comm. Math. Phy. 121, 501–505.
  • M. Campanino, L. Russo (1985), An upper bound on the critical percolation probability for the three-dimensional cubic lattice, Ann. Probab. 13, 478–491.
  • J. Dodziuk (1984), Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284, 787–794.
  • E. Ghys, A. Haefliger and A. Verjovsky eds (1991), Group theory from a geometrical viewpoint, World Scientific.
  • G. R. Grimmett (1989), Percolation, Springer-Verlag, New York.
  • G. R. Grimmett and C. M. Newman (1990), Percolation in $infty+1$ dimensions, in Disorder in physical systems, (G. R. Grimmett and D. J. A. Welsh eds.), Clarendon Press, Oxford, 219–240.
  • O. Häggström (1996), Infinite clusters in dependent automorphism invariant percolation on trees, preprint.
  • T. Hara and G. Slade (1989), The triangle condition for percolation, Bull. Amer. Math. Soc. (N.S.) 21, 269–273.
  • H. Kesten (1980), The critical probability of bond percolation on the square lattice equals ½, Comm. Math. Phys. 74, 41–59.
  • R. Langlands, P. Pouliot, and Y. Saint-Aubin (1994), Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc. (N.S.) 30, 1–61.
  • R. Lyons (1995), Random walks and the growth of groups, C. R. Acad. Sci. Paris Sir. I Math. 320, 1361–1366.
  • R. Lyons (1996), Probability and trees, preprint.
  • W. Magnus, A. Karrass and D. Solitar (1976), Combinatorial group theory, Dover, New York.
  • R. Meester (1994), Uniqueness in percolation theory, Statistica Neerlandica 48 (1994), 237–252.
  • M. V. Men´shikov (1987), Quantitative estimates and strong inequalities for the critical points of a graph and its subgraph, (Russian) Teor. Veroyatnost. i Primenen. 32, 599–602. Eng. transl., Theory Probab. Appl. 32, 544–546.
  • C. M. Newman and L. S. Schulman (1981), Infinite clusters in percolation models, J. Stat. Phys. 26, 613–628.
  • L. Russo (1981), On the critical percolation probabilities, Z. Wahrsch. Verw. Gebiete 56, 229–237.
  • J. C. Wierman (1989), AB Percolation: a brief survey, in Combinatorics and graph theory 25, Banach Center Publications, 241–251.