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arXiv:2407.12603 · Physical review. E

Generalization of the central limit theorem to critical systems: Revisiting perturbation theory

Sankarshan Sahu, Bertrand Delamotte, Adam Rançon

Published Mar 26, 2025• 6 citations
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Abstract

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. 129, 210602 (2022)10.1103/PhysRevLett.129.210602]. It has been shown in that article that there exists an entire family of universal PDFs parameterized by ζ=lim_{L,ξ_{∞}→∞}L/ξ_{∞} which is the ratio of the system size L to the bulk correlation length ξ_{∞} 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.

DOI
10.1103/physreve.111.034128

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Abstract There exists an entire family of universal probability distribution functions (PDFs) of the magnetization mode of the three dimensional Ising model parameterized by ζ = lim L , ξ ∞ L / ξ ∞ which is the ratio of the system size L to the bulk correlation length ξ ∞ with both the thermodynamic limit and the critical limit being taken simultaneously at fixed ζ . Recently, the PDFs of the magnetization mode of the three-dimensional Ising model has been computed at one-loop in the ϵ = 4 − d expansion (2024 arXiv preprint arXiv:2407.12603 ). We show how these PDFs or, equivalently, the rate functions which are their logarithm, can be systematically computed at second order of the perturbative expansion. We compute the whole family of universal rate-functions and show that their agreement with the Monte Carlo data improves significantly at this order when compared to their one-loop counterpart.
We show that the functional renormalization group (FRG) allows for the generalization of the central limit theorem to strongly correlated random variables. On the example of the three-dimensional Ising model at criticality and using the simplest implementation of the FRG, we compute the probability distribution functions of the order parameter or equivalently its logarithm, called the rate functions in large deviations theory. We compute the entire family of universal scaling functions, obtained in the limit where the system size LLL and the correlation length of the infinite system xiinfty\\xi_{\\infty}xiinfty​ diverge, with the ratio zeta=L/xiinfty\\zeta=L/\\xi_{\\infty}zeta=L/xiinfty​ held fixed. It compares very accurately with numerical simulations.
Abstract We show how to compute the probability distributions of the order parameter of the O ( n ) model at two-loop order of perturbation theory generalizing the methods developed for computing the same in case of the Ising model (Sahu 2025 J. Stat. Mech. 2025 123202). We show that even for the O ( n ) model, there exists not one but a family of these probability distribution functions (PDFs) indexed by ζ which is the ratio of system size L to the bulk correlation length ξ ∞ . We also compare these PDFs to the Monte-Carlo simulations and the existing FRG results (Rançon et al 2025 arXiv:2501.04465 [cond-mat.statmech]) for the O (2) and O (3) models.
We study the probability distribution function (PDF) of the order parameter of the three-dimensional O(N) model at criticality using the functional renormalization group. For this purpose, we generalize the method introduced by Balog et al. [Phys. Rev. Lett. 129, 210602 (2022)0031-900710.1103/PhysRevLett.129.210602] to the O(N) model. We study the large N limit, as well as the cases N=2 and N=3 at the level of the local potential approximation (LPA), and compare our results to Monte Carlo simulations. We compute the entire family of universal scaling functions, obtained in the limit where the system size L and the correlation length of the infinite system ξ_{∞} diverge, with the ratio ζ=L/ξ_{∞} constant. We also generalize our results to the approach of criticality from the low-temperature phase where another infinite family of universal PDF exists. We find that the LPA describes very well the functional form of the family of PDFs, once we correct for a global amplitude of the (logarithm of the) PDF and of ζ.
We show that the functional renormalization group (FRG) allows for the calculation of the probability distribution function of the sum of strongly correlated random variables. On the example of the three-dimensional Ising model at criticality and using the simplest implementation of the FRG, we compute the probability distribution functions of the order parameter or, equivalently, its logarithm, called the rate functions in large deviation theory. We compute the entire family of universal scaling functions, obtained in the limit where the system size L and the correlation length of the infinite system ξ_{∞} diverge, with the ratio ζ=L/ξ_{∞} held fixed. It compares very accurately with numerical simulations.
Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.