Skip to main content
SpringerLink
Log in

Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems

  • Chapter
Probabilistic Methods for Algorithmic Discrete Mathematics

Part of the book series: Algorithms and Combinatorics ((AC,volume 16))

  • 2043 Accesses

Abstract

In 1961 Harry Frisch, John Hammersley and I [13] carried out what were in those days massive Monte Carlo experiments attempting to determine the critical percolation probabilities of the various standard lattices. The constraints at that time were, as today, machine induced. The programmes were written in machine code on a computer which was the size of a large room with less power than a modern day calculator. Today the situation has radically changed. Several of these critical probabilities which we were trying to estimate are now known exactly. However the problems posed then have been replaced by problems of just as much charm and seeming intractability and it is some of these that I shall address in these lectures.

Supported in part by Esprit Working Group No. 21726, “RAND2”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alon N., Frieze A.M. and Welsh D.J.A. (1995): Polynomial time randomised approximation schemes for Tutte-Grothendieck invariants: the dense case, Random Structures and Algorithms, 6, 459–478.

    Article  MathSciNet  MATH  Google Scholar 

  2. Annan J D (1994): A randomised approximation algorithm for counting the number of forests in dense graphs, Combinatorics, Probability and Computing, 3, 273–283.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartels E., Mount J. and Welsh D.J.A. (1997): The win polytope of a graph, Annals of Combinatorics 1, 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  4. Björner A., Lovâsz L. and Shor P. (1991): Chip-firing games on graphs, European Journal of Combinatorics 12, 283–291.

    MathSciNet  MATH  Google Scholar 

  5. Broadbent S.R. and Hammersley J.M. (1957): Percolation processes I. Crystals and mazes, Proceedings of the Cambridge Philosophical Society 53, 629–641.

    Article  MathSciNet  MATH  Google Scholar 

  6. Brylawski T.H. and Oxley J.G. (1992): The Tutte polynomial and its applications, Matroid Applications (ed. N. White ), Cambridge Univ. Press, 123–225.

    Google Scholar 

  7. Percolation and the Random Cluster Model 193

    Google Scholar 

  8. Diaz R. and Robins S. (1996): The Ehrhart polynomial of a lattice n-simplex, Electronic Research Announcements of the American Mathematical Society, 2 (1), 1–6.

    Article  MathSciNet  Google Scholar 

  9. Edwards R.G. and Sokal A.D. (1988): Generalization of the Fortuin-KasteleynSwendsen-Wang representation and Monte Carlo algorithms, Phys. Rev. D 38, 2009–2012.

    Article  MathSciNet  Google Scholar 

  10. Ehrhart E. (1967): Sur un problème de géometrie diophantienne linéaire I, II, Journal für die Reine und Angewandte Mathematik, 226, 1–29, and 227, 25–49. Correction 231 (1968), 220.

    Google Scholar 

  11. Essam J.W. and SykesM.F. (1964): Exact critical percolation probabilities for site and bond problems in two dimensions, J. Math. Phys. 5, 1117–1127.

    Article  MathSciNet  Google Scholar 

  12. Fortuin C.M. and Kasteleyn P.W. (1972): On the random cluster model. I Introduction and relation to other models, Physica 57, 536–564.

    Article  MathSciNet  Google Scholar 

  13. Fortuin C.M., Kasteleyn P.W. and Ginibre J. (1971): Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22, 89–103.

    Article  MathSciNet  MATH  Google Scholar 

  14. Frisch H.L., Hammersley J.M. and Welsh D.J.A. (1962), Monte-Carlo estimates of percolation probabilities for various lattices, Physical Review 126, 949–951.

    Article  Google Scholar 

  15. Garey M.R., and Johnson D.S. (1979): Computers and Intractability — A guide to the theory of NP-completeness, W.H. Freeman, San Francisco.

    Google Scholar 

  16. Grimmett G.R. (1989): Percolation, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  17. Grimmett G.R. (1995): The stochastic random-cluster process and the uniqueness of random cluster measures, Annals of Probability 23, 1461–1510.

    Article  MathSciNet  MATH  Google Scholar 

  18. Grimmett G.R. (1997): Percolation and disordered systems, Ecole d’Eté de Probabilités de Saint Flour XXVI–1996 (P. Bernard, ed.) Lecture Notes in Mathematics 1665, Springer-Verlag, Berlin, pp. 153–300.

    Google Scholar 

  19. Grimmett G.R. and Stacey A.M. (1998): Critical probabilities for site and bond percolation models, preprint.

    Google Scholar 

  20. Hammersley J.M. (1961): Comparison of atom and bond percolation, Journal of Mathematical Physics 2, 728–733.

    Article  MathSciNet  MATH  Google Scholar 

  21. Hinterman A., Kunz H. and Wu F.Y. (1978): Exact results for the Potts model in two dimensions, J. Statist. Phys. 19, 623–632.

    Article  MathSciNet  Google Scholar 

  22. Holley R. (1974): Remarks on the FKG inequalities, Comm. Math. Phys. 36, 227–231.

    Article  MathSciNet  Google Scholar 

  23. Jaeger F., Vertigan D.L. and Welsh D.J.A. (1990): On the computational complexity of the Jones and Tutte polynomials, Math. Proc. Camb. Phil. Soc. 108, 35–53.

    Article  MathSciNet  MATH  Google Scholar 

  24. Jerrum M.R. and Sinclair A. (1990): Polynomial-time approximation algorithms for the Ising model, Proc. 17th 1CALP, EATCS, 462–475.

    Google Scholar 

  25. Jerrum M.R., Valiant L.G. and Vazirani V.V. (1986): Random generation of combinatorial structures from a uniform distribution, Theoretical Computer Science 43, 169–188.

    Article  MathSciNet  MATH  Google Scholar 

  26. Karger D.R. (1995): A randomised fully polynomial time approximation scheme for the all terminal network reliability problem, in Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 328–337.

    Google Scholar 

  27. Kasteleyn P.W. (1961): The statistics of dimers on a lattice, Physica 27, 1209 1225.

    Google Scholar 

  28. Kesten H. (1980): The critical probability of bond percolation on the square lattice equals 1/2, Comm. Math. Phys. 74, 41–59.

    Article  MathSciNet  MATH  Google Scholar 

  29. Kesten H. (1982): Percolation Theory for Mathematicians, Birkhauser, Boston (1982).

    Google Scholar 

  30. Laanit L., Messager A., Miracle-Soles S., Ruiz J. and Shlosman S. (1991): Interface in Potts model 1: Pirogov-Sanai theory of the Fortuin-Kasteleyn representation, Comm. Math. Phy. 140, 81–92.

    Article  Google Scholar 

  31. Dominic Welsh

    Google Scholar 

  32. Merino-Lopez C. (1997): Chip-firing and the Tutte polynomial, Annals of Combinatorics 1, 253–259.

    Article  MathSciNet  Google Scholar 

  33. Oxley J.G. and Welsh D.J.A. (1979): The Tutte polynomial and percolation, Graph Theory and Related Topics (eds. J.A. Bondy and U.S.R. Murty), Academic Press, London, 329–339.

    Google Scholar 

  34. Potts R.B. (1952): Some generalised order-disorder transformations, Proceedings Cambridge Philosophical Society 48, 106–109.

    Article  MathSciNet  MATH  Google Scholar 

  35. Stanley R.P. (1980): Decompositions of rational convex polytopes, Annals of Discrete Mathematics 6 (1980), 333–342.

    Article  MathSciNet  Google Scholar 

  36. Swendsen R.H. and Wang J.-S. (1987): Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 86–88.

    Article  Google Scholar 

  37. Thistlethwaite M.B. (1987) A spanning tree expansion of the Jones polynomial, Topology 26, 297–309.

    Article  MathSciNet  MATH  Google Scholar 

  38. Vertigan D.L. and Welsh D.J.A. (1992): The computational complexity of the Tutte plane: the bipartite case. Probability, Combinatorics and Computer Science, 1, 181–187

    Article  MathSciNet  MATH  Google Scholar 

  39. Welsh D.J.A. (1993a): Complexity: Knots, Colourings and Counting, London Mathematical Society Lecture Note Series 186, Cambridge University Press.

    Google Scholar 

  40. Welsh D.J.A. (1993b): Randomised approximation in the Tutte plane, Combinatorics, Probability and Computing, 3, 137–143.

    Article  MathSciNet  Google Scholar 

  41. Welsh D.J.A. (1993c): Percolation in the random cluster model and Q-state Potts model, J. Phys. A (Math. and General) 26, 2471–2483.

    Article  MathSciNet  MATH  Google Scholar 

  42. Welsh D.J.A. (1996): Counting, colourings and flows in random graphs, Bolyai Society Mathematical Studies 2, pp. 491–506.

    MathSciNet  Google Scholar 

  43. Wierman J.C. (1981): Bond percolation on honeycomb and triangular lattices, Adv. Appl. Probab. 13, 293–313.

    MathSciNet  Google Scholar 

  44. Wu F. (1982): The Potts model, Rev. Modern Phys. 54, 235–268.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Oxford, UK

    Dominic Welsh

Authors
  1. Dominic Welsh
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LIRMM, 161, rue Ada, 34392, Montpellier Cedex 5, France

    Michel Habib

  2. Department of Statistics, University of Oxford, 1 South Parks Road, OX1 3TG, Oxford, UK

    Colin McDiarmid

  3. Equipe Combinatoire, Université Pierre et Marie Curie, Paris 6 Case 189, 4, place Jussieu, 75252, Paris Cedex 5, France

    Jorge Ramirez-Alfonsin  & Bruce Reed  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Welsh, D. (1998). Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems. In: Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B. (eds) Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms and Combinatorics, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12788-9_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-12788-9_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08426-3

  • Online ISBN: 978-3-662-12788-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation