The Potts model and the Tutte polynomial

Journal of Mathematical Physics

Volumn 41, Issue 3, 2000, Pages 1127-1152

  Welsh, D J A ^a   Merino, C ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034345415     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533181     Document Type: Review

References (67)

1. D. J. A. Welsh, Complexity: Knots, Colourings and Counting, in London Mathematical Society Lecture Note Series, Vol. 186 (Cambridge University Press, Cambridge, 1993).
2. F. Wu, "The Potts model," Rev. Mod. Phys. 54, 235-268 (1982).
3. J. G. Oxley, Matroid Theory (Oxford University Press, Oxford, 1992).
4. J. G. Oxley and D. J. A. Welsh, "The Tutte polynomial and percolation," Graph Theory and Related Topics, edited by J. A. Bondy and U.S.R. Murty (Academic, London, 1979), pp. 329-339.
5. C. M. Fortuin and P. W. Kasteleyn, "On the random cluster model. I. Introduction and relation to other models," Physica (Amsterdam) 57, 536-564 (1972).
6. G. R. Grimmett, Precolation (Springer, Berlin, 1989).
7. G. R. Grimmett, Precolation and Disordered Systems, Ecole dE'eé de probabilités de Saint Flour XXXVI-1996, edited by P. Bernard [Lect. Notes Math. 1665, 153-300 (1997)].
8. G. R. Grimmett, Precolation and Disordered Systems, Ecole dE'eé de probabilités de Saint Flour XXXVI-1996, edited by P. Bernard [Lect. Notes Math. 1665, 153-300 (1997)].
9. R. G. Edwards and A. D. Sokal, "Generalization of the Fortuin-Kasteleyn-Swendesen-Wang representation and Monte Carlo algorithms," Phys. Rev. D 38, 2009-2012 (1988).
10. R. H. Swendsen and J.-S. Wang, "Nonuniversal critical dynamics in Monte Carlo simulations," Phys. Rev. Lett. 58, 86-88 (1987).
11. R. P. Stanley, "Acyclic orientations of graphs," Discrete Math. 5, 171-178 (1973).
12. T. Zaslavsky, "Facing up to arrangements: Face count formulas for partitions of space by hyperplanes," Memoirs of American Mathematical Society 154 (1975).
13. R. C. Read and P. Rosenstiehl, "On the principal edge tripartition of a graph," Ann. Discrete Math. 3, 195-226 (1978).
14. M. B. Thistlethwaite, "A spanning tree expansion of the Jones polynomial," Topology 26, 297-309 (1987).
15. T. H. Brylawski and J. G. Oxley, The Tutte Polynomial and its Applications Matroid Applications, edited by N. White (Cambridge University Press, Cambridge, 1992), pp. 123-225.
16. H. Kesten and R. H. Schonmann, "Behavior in large dimensions of the Potts and Heisenberg models," Rev. Math. Phys. 1, 147-182 (1989).
17. B. Bollobás, G. Grimmett, and S. Janson, "The random-cluster model on the complete graph," Probab. Theory Related Fields 104, 283-317 (1996).
18. A. Rényi, "Some remarks on the theory of trees," Pub. Math. Inst. Hungarian Acad. Sci. 4, 73-85 (1959).
19. J. Dénes, "The representation of a permutation as the product of a minimal number of transpositions, and its connection with the theory of graphs," Publ. Math. Inst. Hungarian Acad. Sci. 4, 63-71 (1959).
20. W. T. Tutte, "On dichromatic polynomials," J. Comb. Theory, Sec. A 2, 301-320 (1967).
21. D. J. A. Welsh, "Counting, colourings and flows in random graphs," Bolyai Society Mathematical Studies 2, 491-506 (1996).
22. I. M. Gessel and B. E. Sagan, "The Tutte polynomial of a graph depth-first search, and simplicial complex partitions," Electron. J. Combin. 3, 1-36 (1996).
23. L. Takacs, "On the number of distinct forests," SIAM J. Discrete Math. 3, 574-581 (1990).
24. R. P. Stanley, A Zonotope Associated with Graphical Degree Sequences, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 4, 555-570 (1991).
25. F. Jaeger, D. L. Vertigan, and D. J. A. Welsh, "On the computational complexity of the Jones and Tutte polynomials," Math. Proc. Cambridge Philos. Soc. 108, 35-53 (1990).
26. P. W. Kasteleyn, "The statistics of dimers on a lattice," Physica (Amsterdam) 27, 1209-1225 (1961).
27. D. L. Vertigan and D. J. A. Welsh, "The computational complexity of the Tutte plane: The bipartite case," Comb. Probab. Comput. 1, 181-187 (1992).
28. M. R. Jerrum, L. G. Valiant, and V. V. Vazirani, "Random generation of combinatorial structures from a uniform distribution," Theor. Comput. Sci. 43, 169-188 (1986).
29. M. R. Jerrum and A. Sinclair, "Polynomial-time approximation algorithms for the Ising model," SIAM J. Comput. 22, 1087-1116 (1993).
30. D. J. A. Welsh, "Randomised approximation in the Tutte plane," Comb. Probab. Comput. 3, 137-143 (1993).
31. M. Mihail and P. Winkler, "On the number of Eulerian orientations of a graph," Algorithmica 16, 402-414 (1996).
32. J. D. Annan, "A randomized approximation algorithm for counting the number of forests in dense graphs," Comb. Probab. Comput. 3, 273-283 (1994).
33. N. Alon, A. M. Frieze, and D. J. A. Welsh, "Polynomial time randomized approximation schemes for Tutte-Grothendieck invariants: The dense case," Random Struct. Algorithms 6, 459-478 (1995).
34. D. R. Karger, "A randomized 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, 1995 (unpublished), pp. 328-337.
35. C. Hoede, "The Ising model and linear codes," Congr. Numer. 33, 81-108 (1981).
36. A. Rosengren and B. Lindström, "A combinatorial series expansion for the Ising model," Eur. J. Combin. 8, 317-323 (1987).
37. C. Greene, "Weight enumeration and the geometry of linear codes," Stud. Appl. Math. 55, 119-128 (1976).
38. D. J. A. Welsh, Matroid Theory (Academic, New York, 1976).
39. E. Ehrhart, "Sur un problème de géométrie diophantienne linéaire I, II," J. Reine Angew. Math. 226, 1-29 (1967); 227, 25-49 (1967): correction 231, 220 (1968).
40. E. Ehrhart, "Sur un problème de géométrie diophantienne linéaire I, II," J. Reine Angew. Math. 226, 1-29 (1967); 227, 25-49 (1967): correction 231, 220 (1968).
41. R. Diaz and S. Robins, "The Ehrhart, polynomial of a lattice n-simplex," Electron. Res. Announc. Am. Math. Soc. 2, 1-6 (1996).
42. R. P. Stanley, "Decompositions of rational convex polytopes," Ann. Discrete Math. 6, 333-342 (1980).
43. P. Bak, C. Tang, and K. Wiesenfeld, "Self-organized criticality," Phys. Rev. A 38, 364-374 (1988).
44. D. Dhar, "Self-organized critical state of sandpile automaton models," Phys. Rev. Lett. 64, 1613-1616 (1990).
45. A. Gabrielov, "Abelian avalanches and the Tutte polynomials," Physica A 195, 253-274 (1993).
46. A. Björner, L. Lovász, and P. Shor. "Chip-firing games on graphs," Eu. J. Combin. 12, 283-291 (1991).
47. N. Biggs, "The Tutte polynomial as a growth function" (unpublished).
48. C. Merino, "Chip-firing and the Tutte polynomial," Ann. Comb. 1, 253-259 (1997).
49. R. Cori and D. Rossin, "On the sandpile group of a graph" (unpublished).
50. N. Robertson and P. D. Seymour, "Graph Minors. II. Algorithmic aspects of tree-width," J. Algorithm 7, 309-322 (1986).
51. A. Andrzejak, "An algorithm for the Tutte polynomials of graphs of bounded tree width," Discrete Math. 190, 39-54 (1998).
52. S. Noble, "Evaluating the Tutte polynomial for graphs of bounded tree-width," Comb. Probab. Comput. 7, 307-322 (1998).
53. C. Merino and D. J. A. Welsh, "Forests, colorings and acyclic orientations of the square lattice," Ann. Comb. 3, 417-429 (1999).
54. G. R. Grimmett, "Multidimensional lattices and their partition function," Q. J. Math. 29, 141-157 (1978).
55. G. R. Grimmett, "The rank polynomials of large random lattices," J. London Math. Soc. 18, 567-575 (1978).
56. N. L. Bíggs, "Chromatic and thermodynamic limits," J. Phys. A 8, L110-112 (1975).
57. C. J. Thompson, Mathematical Statistical Mechanics (Princeton University Press, Princeton, NJ, 1979), pp. 131-135.
58. N. L. Biggs, "Colouring square lattice graphs," Bull. London Math. Soc. 9, 54-56 (1977).
59. E. Lieb, "The residual entropy of square ice," Phys. Rev. 162, 162-172 (1967).
60. N. L. Biggs and G. H. J. Meredith, "Approximations for chromatic polynomials," J. Comb. Theory, Ser. B 20, 5-19 (1976).
61. J. F. Nagle, "A new sugraph expansion for obtaining coloring polynomials for graphs," J. Comb. Theory, Ser. B 10, 42-59 (1971).
62. D. Kim and I. G. Enting, "The limit of chromatic polynomials," J. Comb. Theory, Ser. B 26, 327-336 (1979).
63. L. Lovász, Combinatorial Problems and Exercises (North-Holland, Amsterdam, 1979), pp. 238-243.
64. H. N. V. Temperley, Combinatorics, in London Mathematical Society, Lecture Note Series Vol. 13 (Cambridge University Press, Cambridge 1974), pp. 202-204.
65. N. J. Calkin, C. Merino, S. Noble, and D. J. A. Welsh (private communication).
66. C. Merino and M. Noy (private communication).
67. E. Bartels, J. Mount, and D. J. A. Welsh, "The win polytope of a graph," Ann. Comb. 1, 1-15 (1997).