Bayesian Inference in the Scaling Analysis of Critical Phenomena

To determine the universality class of critical phenomena, we propose a method of statistical inference in the scaling analysis of critical phenomena. The method is based on Bayesian statistics, most specifically, the Gaussian process regression. It assumes only the smoothness of a scaling function, and it does not need a form. We demonstrate this method for the finite-size scaling analysis of the Ising models on square and triangular lattices. Near the critical point, the method is comparable in accuracy to the least-square method. In addition, it works well for data to which we cannot apply the least-square method with a polynomial of low degree. By comparing the data on triangular lattices with the scaling function inferred from the data on square lattices, we confirm the universality of the finite-size scaling function of the two-dimensional Ising model.

💡 Research Summary

The paper introduces a Bayesian framework for scaling analysis of critical phenomena, focusing on the determination of universality classes without assuming a specific functional form for the scaling function. Traditional scaling analyses rely on least‑square fitting of data to a presumed polynomial form of the scaling function Ψ, which can lead to over‑fitting, requires careful selection of polynomial degree, and is often limited to a narrow region near the critical point. To overcome these limitations, the authors propose using Bayesian inference, specifically Gaussian Process (GP) regression, which only assumes smoothness of Ψ and treats it as a stochastic function characterized by a kernel.

The methodology begins by rescaling raw observables A(t, h) into coordinates (X_i, Y_i) according to the scaling law A = t^x Ψ(ht^{−y}). The statistical error of Y_i is modeled as a multivariate Gaussian with covariance matrix E (diagonal, containing measurement variances). The prior over the unknown scaling function Ψ is expressed as a Gaussian Process with hyper‑parameters θ_h that control the kernel K(X_i, X_j). By choosing a Gaussian kernel K_G(X_i, X_j)=θ_0² exp