Journal of Statistical Physics

, Volume 164, Issue 5, pp 1082–1102 | Cite as

The Worm Process for the Ising Model is Rapidly Mixing

  • Andrea Collevecchio
  • Timothy M. GaroniEmail author
  • Timothy Hyndman
  • Daniel Tokarev


We prove rapid mixing of the worm process for the zero-field ferromagnetic Ising model, on all finite connected graphs, and at all temperatures. As a corollary, we obtain a fully-polynomial randomized approximation scheme for the Ising susceptibility, and for a certain restriction of the two-point correlation function.


Markov chain Mixing time Ising model Worm algorithm 

Mathematics Subject Classification

82B20 82B80 60J10 



The authors wish to gratefully acknowledge the contributions of Greg Markowsky to the early stages of this project, and to also thank Eren Metin Elçi, Catherine Greenhill and Alan Sokal for insightful comments on an earlier draft. T.G. also gratefully acknowledges discussions of the worm process with many colleagues, particularly Youjin Deng, Catherine Greenhill, Alan Sokal, Boris Svistunov and Ulli Wolff. 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.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Andrea Collevecchio
    • 1
  • Timothy M. Garoni
    • 2
    Email author
  • Timothy Hyndman
    • 1
    • 3
  • Daniel Tokarev
    • 1
  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.ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia

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