Third-order transitions in Ising and Potts models on Watts--Strogatz small-world networks
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914387549224960 |
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| author | Wang, Fangfang Liu, Wei Zhang, Ke He, Yongjian Qi, Kai Tang, Ying Di, Zengru |
| author_facet | Wang, Fangfang Liu, Wei Zhang, Ke He, Yongjian Qi, Kai Tang, Ying Di, Zengru |
| contents | We study third-order transitions in the two-dimensional Ising and Potts model on regular lattices and Watts--Strogatz small-world networks. Cluster observables are used to track post-critical boundary reorganization and pre-critical cluster breakup. For the Ising model, the critical temperature $T_c$ is calibrated independently from Binder-cumulant crossings and susceptibility peaks, whereas for the Potts model on small-world networks it is identified operationally from the dominant critical peak of $\mathrm d\langle P\rangle/\mathrm dT$. The independent and dependent third-order transitions are identified from the isolated-spin peak and the post-critical structural extremum, respectively. For both lattice and small-world topologies, we find the robust ordering $T_{\mathrm{ind}}<T_c<T_{\mathrm{dep}}$. Increasing the rewiring probability shifts all three characteristic temperatures upward and enhances the visibility of the post-critical transition. The effect is especially clear in the Potts model, where perimeter-based observables are more sensitive to multistate boundary fluctuations. The systematic persistence of the characteristic temperature hierarchy across topologies and finite sizes argues against interpreting these features as incidental finite-size irregularities. Instead, our results support their interpretation as genuine third-order transitions whose structural detectability can be amplified by network topology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11522 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Third-order transitions in Ising and Potts models on Watts--Strogatz small-world networks Wang, Fangfang Liu, Wei Zhang, Ke He, Yongjian Qi, Kai Tang, Ying Di, Zengru Statistical Mechanics We study third-order transitions in the two-dimensional Ising and Potts model on regular lattices and Watts--Strogatz small-world networks. Cluster observables are used to track post-critical boundary reorganization and pre-critical cluster breakup. For the Ising model, the critical temperature $T_c$ is calibrated independently from Binder-cumulant crossings and susceptibility peaks, whereas for the Potts model on small-world networks it is identified operationally from the dominant critical peak of $\mathrm d\langle P\rangle/\mathrm dT$. The independent and dependent third-order transitions are identified from the isolated-spin peak and the post-critical structural extremum, respectively. For both lattice and small-world topologies, we find the robust ordering $T_{\mathrm{ind}}<T_c<T_{\mathrm{dep}}$. Increasing the rewiring probability shifts all three characteristic temperatures upward and enhances the visibility of the post-critical transition. The effect is especially clear in the Potts model, where perimeter-based observables are more sensitive to multistate boundary fluctuations. The systematic persistence of the characteristic temperature hierarchy across topologies and finite sizes argues against interpreting these features as incidental finite-size irregularities. Instead, our results support their interpretation as genuine third-order transitions whose structural detectability can be amplified by network topology. |
| title | Third-order transitions in Ising and Potts models on Watts--Strogatz small-world networks |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2603.11522 |