Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems

  • Dominic Welsh
Part of the Algorithms and Combinatorics book series (AC, volume 16)


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 However the problems posed then have been replaced by problems of just as much charm and seeming intractability and it is some of these that I shall address in these lectures.


Partition Function Ising Model Critical Probability Jones Polynomial Weight Enumerator 
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  1. 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.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bartels E., Mount J. and Welsh D.J.A. (1997): The win polytope of a graph, Annals of Combinatorics 1, 1–15.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Björner A., Lovâsz L. and Shor P. (1991): Chip-firing games on graphs, European Journal of Combinatorics 12, 283–291.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Broadbent S.R. and Hammersley J.M. (1957): Percolation processes I. Crystals and mazes, Proceedings of the Cambridge Philosophical Society 53, 629–641.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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 193Google Scholar
  8. 7.
    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.MathSciNetCrossRefGoogle Scholar
  9. 8.
    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.MathSciNetCrossRefGoogle Scholar
  10. 9.
    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. 10.
    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.MathSciNetCrossRefGoogle Scholar
  12. 11.
    Fortuin C.M. and Kasteleyn P.W. (1972): On the random cluster model. I Introduction and relation to other models, Physica 57, 536–564.MathSciNetCrossRefGoogle Scholar
  13. 12.
    Fortuin C.M., Kasteleyn P.W. and Ginibre J. (1971): Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22, 89–103.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 13.
    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.CrossRefGoogle Scholar
  15. 14.
    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. 15.
    Grimmett G.R. (1989): Percolation, Springer-Verlag, Berlin.zbMATHGoogle Scholar
  17. 16.
    Grimmett G.R. (1995): The stochastic random-cluster process and the uniqueness of random cluster measures, Annals of Probability 23, 1461–1510.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 17.
    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. 18.
    Grimmett G.R. and Stacey A.M. (1998): Critical probabilities for site and bond percolation models, preprint.Google Scholar
  20. 19.
    Hammersley J.M. (1961): Comparison of atom and bond percolation, Journal of Mathematical Physics 2, 728–733.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 20.
    Hinterman A., Kunz H. and Wu F.Y. (1978): Exact results for the Potts model in two dimensions, J. Statist. Phys. 19, 623–632.MathSciNetCrossRefGoogle Scholar
  22. 21.
    Holley R. (1974): Remarks on the FKG inequalities, Comm. Math. Phys. 36, 227–231.MathSciNetCrossRefGoogle Scholar
  23. 22.
    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.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 23.
    Jerrum M.R. and Sinclair A. (1990): Polynomial-time approximation algorithms for the Ising model, Proc. 17th 1CALP, EATCS, 462–475.Google Scholar
  25. 24.
    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.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 25.
    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. 26.
    Kasteleyn P.W. (1961): The statistics of dimers on a lattice, Physica 27, 1209 1225.Google Scholar
  28. 27.
    Kesten H. (1980): The critical probability of bond percolation on the square lattice equals 1/2, Comm. Math. Phys. 74, 41–59.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 28.
    Kesten H. (1982): Percolation Theory for Mathematicians, Birkhauser, Boston (1982).Google Scholar
  30. 29.
    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.CrossRefGoogle Scholar
  31. 194.
    Dominic WelshGoogle Scholar
  32. 30.
    Merino-Lopez C. (1997): Chip-firing and the Tutte polynomial, Annals of Combinatorics 1, 253–259.MathSciNetCrossRefGoogle Scholar
  33. 31.
    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. 32.
    Potts R.B. (1952): Some generalised order-disorder transformations, Proceedings Cambridge Philosophical Society 48, 106–109.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 33.
    Stanley R.P. (1980): Decompositions of rational convex polytopes, Annals of Discrete Mathematics 6 (1980), 333–342.MathSciNetCrossRefGoogle Scholar
  36. 34.
    Swendsen R.H. and Wang J.-S. (1987): Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 86–88.CrossRefGoogle Scholar
  37. 35.
    Thistlethwaite M.B. (1987) A spanning tree expansion of the Jones polynomial, Topology 26, 297–309.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 36.
    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, 181–187MathSciNetzbMATHCrossRefGoogle Scholar
  39. 37.
    Welsh D.J.A. (1993a): Complexity: Knots, Colourings and Counting, London Mathematical Society Lecture Note Series 186, Cambridge University Press.Google Scholar
  40. 38.
    Welsh D.J.A. (1993b): Randomised approximation in the Tutte plane, Combinatorics, Probability and Computing, 3, 137–143.MathSciNetCrossRefGoogle Scholar
  41. 39.
    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.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 40.
    Welsh D.J.A. (1996): Counting, colourings and flows in random graphs, Bolyai Society Mathematical Studies 2, pp. 491–506.MathSciNetGoogle Scholar
  43. 41.
    Wierman J.C. (1981): Bond percolation on honeycomb and triangular lattices, Adv. Appl. Probab. 13, 293–313.MathSciNetGoogle Scholar
  44. 42.
    Wu F. (1982): The Potts model, Rev. Modern Phys. 54, 235–268.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Dominic Welsh
    • 1
  1. 1.University of OxfordUK

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