Some geometric critical exponents for percolation and the random-cluster model

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 81, Issue 2, 2010, Pages

  Deng, Youjin ^a   Zhang, Wei ^b   Garoni, Timothy M ^c   Sokal, Alan D ^d,e   Sportiello, Andrea ^f  

^a UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

^b JINAN UNIVERSITY   (China)

^c UNIVERSITY OF MELBOURNE   (Australia)

^d POLYTECHNIC UNIVERSITY   (United States)

^e UNIVERSITY COLLEGE LONDON   (United Kingdom)

^f UNIVERSITY OF MILAN   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

CRITICAL EXPONENT; MONTE CARLO ALGORITHMS; MONTE CARLO DATA; MONTE CARLO SIMULATION; RANDOM-CLUSTER MODEL; SHORTEST-PATH; TWO-DIMENSION;

COMPUTER SIMULATION; FRACTAL DIMENSION; GENE EXPRESSION; SOLVENTS;

MONTE CARLO METHODS;

EID: 77249148424     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.81.020102     Document Type: Article

References (41)

1. P. W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. Jpn. 26 (Suppl.), 11 (1969)
2. C. M. Fortuin and P. W. Kasteleyn, Physica 57, 536 (1972) 10.1016/0031-8914(72)90045-6
3. G. Grimmett, The Random-Cluster Model (Springer, New York, 2006).
4. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer, New York, 1997).
5. O. Schramm, Isr. J. Math. 118, 221 (2000) 10.1007/BF02803524
6. S. Rohde and O. Schramm, Ann. Math. 161, 883 (2005) 10.4007/annals.2005. 161.883
7. G. F. Lawler, Conformally Invariant Processes in the Plane (American Mathematical Society, Providence, 2005).
8. W. Kager and B. Nienhuis, J. Stat. Phys. 115, 1149 (2004) 10.1023/B:JOSS.0000028058.87266.be
9. J. Cardy, Ann. Phys. 318, 81 (2005). 10.1016/j.aop.2005.04.001
10. D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor & Francis, London, 1992)
11. G. Grimmett, Percolation, 2nd ed. (Springer, Berlin, 1999).
12. R. B. Potts, Proc. Cambridge Philos. Soc. 48, 106 (1952) 10.1017/S0305004100027419
13. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982) 10.1103/RevModPhys.54.235
14. F. Y. Wu, Rev. Mod. Phys. 55, 315 (E) (1983) 10.1103/RevModPhys.55.315
15. F. Y. Wu, J. Appl. Phys. 55, 2421 (1984). 10.1063/1.333682
16. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982).
17. R. J. Baxter, S. B. Kelland, and F. Y. Wu, J. Phys. A 9, 397 (1976). 10.1088/0305-4470/9/3/009
18. H. Saleur and B. Duplantier, Phys. Rev. Lett. 58, 2325 (1987). 10.1103/PhysRevLett.58.2325
19. H. J. Herrmann, D. C. Hong, and H. E. Stanley, J. Phys. A 17, L261 (1984). 10.1088/0305-4470/17/5/008
20. B. Nienhuis, J. Stat. Phys. 34, 731 (1984). 10.1007/BF01009437
21. M. Aizenman, B. Duplantier, and A. Aharony, Phys. Rev. Lett. 83, 1359 (1999) 10.1103/PhysRevLett.83.1359
22. H. Kesten, Commun. Math. Phys. 109, 109 (1987) 10.1007/BF01205674
23. S. Smirnov and W. Werner, Math. Res. Lett. 8, 729 (2001).
24. A. D. Sokal, in Functional Integration: Basics and Applications, edited by, C. de Witt-Morette, P. Cartier, and A. Folacci, (Plenum, New York, 1997), pp. 131-192.
25. M. Sweeny, Phys. Rev. B 27, 4445 (1983). 10.1103/PhysRevB.27.4445
26. Y. Deng, T. M. Garoni, and A. D. Sokal, Phys. Rev. Lett. 98, 230602 (2007). 10.1103/PhysRevLett.98.230602
27. X.-J. Li and A. D. Sokal, Phys. Rev. Lett. 63, 827 (1989), see last paragraph. 10.1103/PhysRevLett.63.827
28. M. R. Henzinger and V. King, J. ACM 46, 502 (1999) 10.1145/320211.320215
29. J. Holm, K. de Lichtenberg, and M. Thorup, J. ACM 48, 723 (2001). 10.1145/502090.502095
30. - x 2 to the total cost.
31. S. S. Manna, D. Dhar, and S. N. Majumdar, Phys. Rev. A 46, R4471 (1992). 10.1103/PhysRevA.46.R4471
32. J. Cardy, J. Phys. A 31, L105 (1998). 10.1088/0305-4470/31/5/003
33. R. M. Ziff, J. Phys. A 32, L457 (1999). 10.1088/0305-4470/32/43/101
34. We avoided q = 4 because the logarithmic corrections would likely make all exponent estimates unreliable [see, e.g., J. Salas and A. D. Sokal, J. Stat. Phys. 88, 567 (1997).] 10.1023/B:JOSS.0000015164.98296.85
35. L. Chayes and J. Machta, Physica A 254, 477 (1998) 10.1016/S0378-4371(97) 00637-7
36. Y. Deng, et al. Phys. Rev. Lett. 99, 055701 (2007). 10.1103/PhysRevLett. 99.055701
37. D. B. Wilson, in Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (ACM, New York, 1996), pp. 296-303.
38. More precisely, we used Wilson's algorithm to generate a random spanning tree T, and then took measurements on T \ T 0, where T 0 is a fixed short tree connecting x 1,..., x k.
39. See supplementary material at http://link.aps.org/supplemental/10.1103/ PhysRevE.81.020102 for full tables.
40. P. Grassberger, J. Phys. A 32, 6233 (1999). 10.1088/0305-4470/32/35/301
41. Y. Deng, H. W. J. Blöte, and B. Nienhuis, Phys. Rev. E 69, 026114 (2004). Note that 4 ≤ g ≤ 6 corresponds to the tricritical Potts model. 10.1103/PhysRevE.69.026114