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Dynamic Critical Behavior of the Chayes–Machta Algorithm for the Random-Cluster Model, I. Two Dimensions

Journal of Statistical Physics volume 144, pages 459–518 (2011)Cite this article

Abstract

We study, via Monte Carlo simulation, the dynamic critical behavior of the Chayes–Machta dynamics for the Fortuin–Kasteleyn random-cluster model, which generalizes the Swendsen–Wang dynamics for the q-state Potts ferromagnet to non-integer q≥1. We consider spatial dimension d=2 and 1.25≤q≤4 in steps of 0.25, on lattices up to 10242, and obtain estimates for the dynamic critical exponent z CM. We present evidence that when 1≤q≲1.95 the Ossola–Sokal conjecture z CM≥β/ν is violated, though we also present plausible fits compatible with this conjecture. We show that the Li–Sokal bound z CM≥α/ν is close to being sharp over the entire range 1≤q≤4, but is probably non-sharp by a power. As a byproduct of our work, we also obtain evidence concerning the corrections to scaling in static observables.

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Author notes

  1. Timothy M. Garoni

    Present address: School of Mathematical Sciences, Monash University, Melbourne, Victoria, 3800, Australia

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Melbourne, Melbourne, Victoria, 3010, Australia

    Timothy M. Garoni

  2. Department of Physics, New York City College of Technology, 300 Jay Street, Brooklyn, NY, 11201, USA

    Giovanni Ossola

  3. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Marco Polin

  4. Department of Physics, New York University, 4 Washington Place, New York, NY, 10003, USA

    Alan D. Sokal

  5. Department of Mathematics, University College London, London, WC1E 6BT, UK

    Alan D. Sokal

Authors
  1. Timothy M. Garoni
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  2. Giovanni Ossola
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  3. Marco Polin
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  4. Alan D. Sokal
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Corresponding author

Correspondence to Alan D. Sokal.

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Garoni, T.M., Ossola, G., Polin, M. et al. Dynamic Critical Behavior of the Chayes–Machta Algorithm for the Random-Cluster Model, I. Two Dimensions. J Stat Phys 144, 459–518 (2011). https://doi.org/10.1007/s10955-011-0267-y

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  • DOI: https://doi.org/10.1007/s10955-011-0267-y

Keywords

  • Random-cluster model
  • Potts model
  • Chayes–Machta algorithm
  • Swendsen–Wang algorithm
  • Cluster algorithm
  • Dynamic critical behavior