Sweeny dynamics for the random-cluster model with small $Q$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929252047257600 |
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| author | Peng, Zirui Elçi, Eren Metin Deng, Youjin Hu, Hao |
| author_facet | Peng, Zirui Elçi, Eren Metin Deng, Youjin Hu, Hao |
| contents | The Sweeny algorithm for the $Q$-state random-cluster model in two dimensions is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As $Q$ decreases, the so-called critical speeding-up for non-local quantities becomes more and more pronounced. However, for some quantity of specific local pattern -- e.g., the number of half faces on the square lattice, we observe that, as $Q \to 0$, the integrated autocorrelation time $τ$ diverges as $Q^{-ζ}$, with $ζ\simeq 1/2$, leading to the non-ergodicity of the Sweeny method for $Q \to 0$. Such $Q$-dependent critical slowing-down, attributed to the peculiar form of the critical bond weight $v=\sqrt{Q}$, can be eliminated by a combination of the Sweeny and the Kawasaki algorithm. Moreover, by classifying the occupied bonds into bridge bonds and backbone bonds, and the empty bonds into internal-perimeter bonds and external-perimeter bonds, one can formulate an improved version of the Sweeny-Kawasaki method such that the autocorrelation time for any quantity is of order $O(1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00254 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sweeny dynamics for the random-cluster model with small $Q$ Peng, Zirui Elçi, Eren Metin Deng, Youjin Hu, Hao Statistical Mechanics The Sweeny algorithm for the $Q$-state random-cluster model in two dimensions is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As $Q$ decreases, the so-called critical speeding-up for non-local quantities becomes more and more pronounced. However, for some quantity of specific local pattern -- e.g., the number of half faces on the square lattice, we observe that, as $Q \to 0$, the integrated autocorrelation time $τ$ diverges as $Q^{-ζ}$, with $ζ\simeq 1/2$, leading to the non-ergodicity of the Sweeny method for $Q \to 0$. Such $Q$-dependent critical slowing-down, attributed to the peculiar form of the critical bond weight $v=\sqrt{Q}$, can be eliminated by a combination of the Sweeny and the Kawasaki algorithm. Moreover, by classifying the occupied bonds into bridge bonds and backbone bonds, and the empty bonds into internal-perimeter bonds and external-perimeter bonds, one can formulate an improved version of the Sweeny-Kawasaki method such that the autocorrelation time for any quantity is of order $O(1)$. |
| title | Sweeny dynamics for the random-cluster model with small $Q$ |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2308.00254 |