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Journal of Statistical Physics

, Volume 170, Issue 1, pp 22–61 | Cite as

On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model

  • Andrea Collevecchio
  • Eren Metin Elçi
  • Timothy M. GaroniEmail author
  • Martin Weigel
Article

Abstract

We consider the coupling from the past implementation of the random–cluster heat-bath process, and study its random running time, or coupling time. We focus on hypercubic lattices embedded on tori, in dimensions one to three, with cluster fugacity at least one. We make a number of conjectures regarding the asymptotic behaviour of the coupling time, motivated by rigorous results in one dimension and Monte Carlo simulations in dimensions two and three. Amongst our findings, we observe that, for generic parameter values, the distribution of the appropriately standardized coupling time converges to a Gumbel distribution, and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time. Perhaps surprisingly, we observe these results to hold both off criticality, where the coupling time closely mimics the coupon collector’s problem, and also at the critical point, provided the cluster fugacity is below the value at which the transition becomes discontinuous. Finally, we consider analogous questions for the single-spin Ising heat-bath process.

Keywords

Coupling from the past Relaxation time Random–cluster model Markov-chain Monte Carlo 

Notes

Acknowledgements

The authors thank Youjin Deng, Alan Sokal, and Ulli Wolff for useful discussions, and an anonymous referee for helpful comments. This work was supported under the Australian Research Council’s Discovery Projects funding scheme (project numbers DP140100559 & DP110101141), and T.G. is the recipient of an Australian Research Council Future Fellowship (project number FT100100494). A.C. would like to thank STREP project MATHEMACS. The work of EE and MW was partially supported by the European Commission through the IRSES network DIONICOS (PIRSES-GA-2013-612707).

References

  1. 1.
    Aizenman, M., Duminil-Copin, H., Sidoravicius, V.: Random currents and continuity of ising model’s spontaneous magnetization. Commun. Math. Phys. 334, 719–742 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baxter, R.J.: Solvable eight-vertex model on an arbitrary planar lattice. Philos. Trans. R. Soc. A 289, 315–346 (1978)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Beffara, V., Duminil-Copin, H.: The self-dual point of the two-dimensional random-cluster model is critical for q \(\ge \) 1. Probab. Theory Relat. Fields 153, 511–542 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Billingsley, P.: Probability and Measure. (Wiley Series in Probability and Statistics), 3rd edn. Wiley, New York (1994)zbMATHGoogle Scholar
  5. 5.
    Cesi, F., Guadagni, G., Martinelli, F., Schonmann, R.H.: On the two-dimensional stochastic Ising model in the phase coexistence region near the critical point. J. Stat. Phys. 85, 55–102 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chayes, L., Machta, J.: Graphical representations and cluster algorithms II. Phys. A 254, 477–516 (1998)CrossRefGoogle Scholar
  7. 7.
    Chow, Y.S., Teicher, H.: Probability Theory: Independence, Interchangeability, Martingales. Springer, New York (1978)CrossRefzbMATHGoogle Scholar
  8. 8.
    Deng, Y., Blöte, H.: Simultaneous analysis of several models in the three-dimensional Ising universality class. Phys. Rev. E 68, 036125 (2003)ADSCrossRefGoogle Scholar
  9. 9.
    Deng, Y., Garoni, T., Machta, J., Ossola, G., Polin, M., Sokal, A.: Critical behavior of the Chayes–Machta–Swendsen–Wang dynamics. Phys. Rev. Lett. 99, 055701 (2007)ADSCrossRefGoogle Scholar
  10. 10.
    Deng, Y., Garoni, T.M., Sokal, A.D.: Critical speeding-up in the local dynamics of the random-cluster model. Phys. Rev. Lett. 98, 230602 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    Duminil-Copin, H., Gagnebin, M., Harel, M., Manolescu, I., Tassion, V.: Discontinuity of the phase transition for the planar random-cluster and Potts models with \(q > 4\). arXiv:1611.09877 (2016)
  12. 12.
    Duminil-Copin, H., Sidoravicius, V., Tassion, V.: Continuity of the phase transition for planar random-cluster and Potts models with \(1 \le q \le 4\). arXiv:1505.04159 (2015)
  13. 13.
    Dyer, M., Greenhill, C., Ullrich, M.: Structure and eigenvalues of heat-bath Markov chains. Linear Algebra Appl. 454, 57–71 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Elci, E.: Algorithmic and geometric aspects of the random-cluster model. Ph.D. thesis (2015)Google Scholar
  15. 15.
    Elçi, E.M., Weigel, M.: Efficient simulation of the random-cluster model. Phys. Rev. E 88, 033303 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    Elçi, E.M., Weigel, M.: Dynamic connectivity algorithms for Monte Carlo simulations of the random-cluster model. J. Phys. 510(1), 012013 (2014)Google Scholar
  17. 17.
    Erdos, P., Renyi, A.: On a classical problem of probability theory. Publ. Math. Inst. Hung. Acad. Sci. Ser. A 6, 215–219 (1961)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)zbMATHGoogle Scholar
  19. 19.
    Friedli, S., Velenik, Y.: Statistical Mechanics of Lattice Systems. A Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2016)zbMATHGoogle Scholar
  20. 20.
    Gheissari, R., Lubetzky, E.: Mixing Times Of Critical 2D Potts Models. arXiv:1607.02182 (2016)
  21. 21.
    Gliozzi, F.: Simulation of Potts models with real q and no critical slowing down. Phys. Rev. E 66, 016115 (2002)ADSCrossRefGoogle Scholar
  22. 22.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. A Foundation for Computer Science, 2nd edn. Addisen-Wesley Publishing Company, Boston (1994)zbMATHGoogle Scholar
  23. 23.
    Grassberger, P.: Damage spreading and critical exponents for “model A” Ising dynamics. Phys. A 214, 547–559 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Grimmett, G.: The Random-Cluster Model. Springer, New York (2006)CrossRefzbMATHGoogle Scholar
  25. 25.
    Grimmett, G.: Probability on Graphs. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  26. 26.
    Guo, H., Jerrum, M.: Random cluster dynamics for the Ising model is rapidly mixing. arXiv:1605.00139 pp. 1–15 (2016)
  27. 27.
    Häggström, O.: Finite Markov Chains and Algorithmic Applications. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  28. 28.
    Hartmann, A.: Calculation of partition functions by measuring component distributions. Phys. Rev. Lett. 94, 050601 (2005)ADSCrossRefGoogle Scholar
  29. 29.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM (JACM) 48, 723–760 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Philos. Soc. 108, 35–53 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Janson, S.: Tail bounds for sums of geometric and exponential random variables (2014). http://www2.math.uu.se/~svante/papers/sjN14.pdf
  32. 32.
    Jerrum, M.: Mathematical Foundations of the Markov Chain Monte Carlo Method. In: Probabilistic Methods for Algorithmic Discrete Mathematics, pp. 116–165. Springer, New York (1998)Google Scholar
  33. 33.
    Laanait, L., Messager, A., Miracle-Solé, S., Ruiz, J., Shlosman, S.: Interfaces in the Potts model I: Pirogov–Sinai theory of the Fortuin–Kasteleyn representation. Commun. Math. Phys. 140(1), 81–91 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and related properties of random sequences and processes. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  35. 35.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  36. 36.
    Madras, N., Slade, G.: The Self-Avoiding Walk. Birkhauser, Boston (1996)CrossRefzbMATHGoogle Scholar
  37. 37.
    McCoy, B.M., Wu, T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)CrossRefzbMATHGoogle Scholar
  38. 38.
    Mitzenmacher, M., Upfal, E.: Probability and Computing. Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  39. 39.
    Nacu, S.: Glauber dynamics on the cycle is monotone. Probab. Theory Relat. Fields 127, 177–185 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Nienhuis, B.: Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys. 34, 731–761 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Nightingale, M.P., Bloete, H.W.J.: Dynamic exponent of the two-dimensional Ising model and Monte Carlo computation of the subdominant eigenvalue of the stochastic matrix. Phys. Rev. Lett. 76, 4548–4551 (1996)ADSCrossRefGoogle Scholar
  42. 42.
    Posfai, A.: Approximation Theorems Related to the Coupon Collector’s Problem . Ph.D. thesis (2010)Google Scholar
  43. 43.
    Propp, J., Wilson, D.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Algorithms 9, 223–252 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Sinclair, A.B., Alistair, Sinclair, A.: Random-Cluster Dynamics in \(\mathbb{Z}^2\). In: Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 498–513 (2016)Google Scholar
  45. 45.
    Sokal, A.D.: Monte Carlo methods in statistical mechanics: foundations and new algorithms. In: DeWitt-Morette, C., Cartier, P., Folacci, A. (eds.) Functional Integration: Basics and Applications (1996 Cargèse summer school), pp. 131–192. Plenum, New York (1997)CrossRefGoogle Scholar
  46. 46.
    Sweeny, M.: Monte Carlo study of weighted percolation clusters relevant to the Potts models. Phys. Rev. B 27, 4445–4455 (1983)ADSCrossRefGoogle Scholar
  47. 47.
    Swendsen, R.H., Wang, J.S.: Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987)ADSCrossRefGoogle Scholar
  48. 48.
    Wang, J.S., Kozan, O., Swendsen, R.: Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation. Phys. Rev. E 66(5), 057101 (2002)ADSCrossRefGoogle Scholar
  49. 49.
    Welsh, D.J.A.: Complexity: Knots, Colourings and Counting, London Mathematical Society Lecture Note Series, vol. 186. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  50. 50.
    Deng, Youjin, Garoni, Timothy, M., Sokal, Alan, Zhou, Zongzheng: Dynamic critical behavior of the Chayes-Machta random-cluster algorithm II: Three-dimensions. In preparationGoogle Scholar
  51. 51.
    Young, P.: Everything You Wanted to Know about Data Analysis and Fitting but were Afraid to ask. Springer, New York (2015)CrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia
  2. 2.ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical SciencesMonash UniversityClaytonAustralia
  3. 3.Applied Mathematics Research CentreCoventry UniversityCoventryUnited Kingdom

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