Percolation in the three-dimensional Ising model
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914451960102912 |
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| author | Zhu, Jinhong Chen, Tao Li, Zhiyi Fang, Sheng Deng, Youjin |
| author_facet | Zhu, Jinhong Chen, Tao Li, Zhiyi Fang, Sheng Deng, Youjin |
| contents | 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 $y_p = 0.426(6)$ and the fractal dimensions of the largest cluster, hull, and shortest path as $d_f = 1.8926(20)$, $d_{\rm hull} = 1.663(4)$, and $d_{\rm min} = 1.080(10)$, respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05772 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Percolation in the three-dimensional Ising model Zhu, Jinhong Chen, Tao Li, Zhiyi Fang, Sheng Deng, Youjin Statistical Mechanics 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 $y_p = 0.426(6)$ and the fractal dimensions of the largest cluster, hull, and shortest path as $d_f = 1.8926(20)$, $d_{\rm hull} = 1.663(4)$, and $d_{\rm min} = 1.080(10)$, respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations. |
| title | Percolation in the three-dimensional Ising model |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2604.05772 |