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

TL;DR

This paper extends the understanding of probability distributions near critical points by systematically computing universal PDFs using perturbation theory, improving agreement with numerical data for critical systems.

Contribution

It introduces a perturbative method to compute the family of universal PDFs for critical systems, connecting renormalization group results with perturbation theory.

Findings

Derived the entire family of universal PDFs using epsilon-expansion.

Showed qualitative agreement with Monte Carlo simulations.

Proposed methods to enhance quantitative accuracy of theoretical predictions.

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 $\zeta=\lim_{L,\xi_\infty\rightarrow\infty} L/\xi_\infty$ 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.