Generalization of the Central Limit Theorem to Critical Systems: Revisiting Perturbation Theory
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909554551291904 |
|---|---|
| author | Sahu, Sankarshan Delamotte, Bertrand Rançon, Adam |
| author_facet | Sahu, Sankarshan Delamotte, Bertrand Rançon, Adam |
| contents | The Central Limit Theorem does not hold for strongly correlated stochastic variables, as is the case for statistical systems close to criticality. Recently, the calculation of the probability distribution function (PDF) of the magnetization mode has been performed with the functional renormalization group in the case of the three-dimensional Ising model [Balog et al., Phys. Rev. Lett. {\bf 129}, 210602 (2022)]. It has been shown in that article that there exists an entire family of universal PDFs parameterized by $ζ=\lim_{L,ξ_\infty\rightarrow\infty} L/ξ_\infty$ which is the ratio of the system size $L$ to the bulk correlation length $ξ_{\infty}$ with both the thermodynamic limit and the critical limit being taken simultaneously. We show how these PDFs or, equivalently, the rate functions which are their logarithm, can be systematically computed perturbatively in the $ε=4-d$ expansion. We determine the whole family of universal PDFs and show that they are in good qualitative agreement with Monte Carlo data. Finally, we conjecture on how to significantly improve the quantitative agreement between the one-loop and the numerical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12603 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalization of the Central Limit Theorem to Critical Systems: Revisiting Perturbation Theory Sahu, Sankarshan Delamotte, Bertrand Rançon, Adam Statistical Mechanics High Energy Physics - Theory The Central Limit Theorem does not hold for strongly correlated stochastic variables, as is the case for statistical systems close to criticality. Recently, the calculation of the probability distribution function (PDF) of the magnetization mode has been performed with the functional renormalization group in the case of the three-dimensional Ising model [Balog et al., Phys. Rev. Lett. {\bf 129}, 210602 (2022)]. It has been shown in that article that there exists an entire family of universal PDFs parameterized by $ζ=\lim_{L,ξ_\infty\rightarrow\infty} L/ξ_\infty$ which is the ratio of the system size $L$ to the bulk correlation length $ξ_{\infty}$ with both the thermodynamic limit and the critical limit being taken simultaneously. We show how these PDFs or, equivalently, the rate functions which are their logarithm, can be systematically computed perturbatively in the $ε=4-d$ expansion. We determine the whole family of universal PDFs and show that they are in good qualitative agreement with Monte Carlo data. Finally, we conjecture on how to significantly improve the quantitative agreement between the one-loop and the numerical results. |
| title | Generalization of the Central Limit Theorem to Critical Systems: Revisiting Perturbation Theory |
| topic | Statistical Mechanics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2407.12603 |