Generalization of the Central Limit Theorem to Critical Systems: Revisiting Perturbation Theory

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. {\bf 129}, 210602 (2022)]. It has been shown in that article that there exists an entire family of universal PDFs parameterized by $\zeta ={\lim }_{L,{\xi }_{\infty }\to \infty }L/{\xi }_{\infty }$ which is the ratio of the system size $L$ to the bulk correlation length ${\xi }_{\infty }$ 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 $\epsilon =4-d$ 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.

v1) 15 pages, 11 figures; v2) minor corrections

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Cited by in corpus (6)

• Universal and non-universal large deviations in critical systems
• Critical Probability Distributions of the order parameter at two loops I: Ising universality class
• Constraint effective action and critical correlation functions at fixed magnetization
• Constraint correlation functions of the one-dimensional Ising model in the scaling limit
• Critical Probability Distributions of the order parameter at two loops II: $O(n)$ universality class
• Observation of universal non-Gaussian statistics of the order parameter across a continuous phase transition