Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913724202221568 |
|---|---|
| author | Xu, Yihao Chen, Tao Zhou, Zongzheng Salas, Jesús Deng, Youjin |
| author_facet | Xu, Yihao Chen, Tao Zhou, Zongzheng Salas, Jesús Deng, Youjin |
| contents | The number $n_s$ of clusters (per site) of size $s$, a central quantity in percolation theory, displays at criticality an algebraic scaling behavior of the form $n_s\simeq s^{-τ}\, A\, (1+B s^{-Ω})$. For the Fortuin--Kasteleyn representation of the $Q$-state Potts model in two dimensions, the Fisher exponent $τ$ is known as a function of the real parameter $0\le Q\le4$, and, for bond percolation (the $Q\rightarrow 1$ limit), the correction-to-scaling exponent is derived as $Ω=72/91$. We theoretically derive the exact formula for the correction-to-scaling exponent $Ω=8/[(2g+1)(2g+3)]$ as a function of the Coulomb-gas coupling strength $g$, which is related to $Q$ by $Q=2+2\cos(2 πg)$. Using an efficient Monte Carlo cluster algorithm, we study the O($n$) loop model on the hexagonal lattice, which is in the same universality class as the $Q=n^2$ Potts model, and has significantly suppressed finite-size corrections and critical slowing-down. The predictions of the above formula include the exact value for percolation as a special case and agree well with the numerical estimates of $Ω$ for both the critical and tricritical branches of the Potts model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12646 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions Xu, Yihao Chen, Tao Zhou, Zongzheng Salas, Jesús Deng, Youjin Statistical Mechanics The number $n_s$ of clusters (per site) of size $s$, a central quantity in percolation theory, displays at criticality an algebraic scaling behavior of the form $n_s\simeq s^{-τ}\, A\, (1+B s^{-Ω})$. For the Fortuin--Kasteleyn representation of the $Q$-state Potts model in two dimensions, the Fisher exponent $τ$ is known as a function of the real parameter $0\le Q\le4$, and, for bond percolation (the $Q\rightarrow 1$ limit), the correction-to-scaling exponent is derived as $Ω=72/91$. We theoretically derive the exact formula for the correction-to-scaling exponent $Ω=8/[(2g+1)(2g+3)]$ as a function of the Coulomb-gas coupling strength $g$, which is related to $Q$ by $Q=2+2\cos(2 πg)$. Using an efficient Monte Carlo cluster algorithm, we study the O($n$) loop model on the hexagonal lattice, which is in the same universality class as the $Q=n^2$ Potts model, and has significantly suppressed finite-size corrections and critical slowing-down. The predictions of the above formula include the exact value for percolation as a special case and agree well with the numerical estimates of $Ω$ for both the critical and tricritical branches of the Potts model. |
| title | Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2411.12646 |