Bayesian Inference in the Scaling Analysis of Critical Phenomena

TL;DR

This paper introduces a Bayesian inference method using Gaussian process regression for scaling analysis of critical phenomena, enabling universality confirmation with minimal assumptions and improved flexibility over traditional methods.

Contribution

It presents a novel Bayesian approach for critical phenomena scaling analysis that requires only smoothness, avoiding specific functional forms, and demonstrates its effectiveness on Ising models.

Findings

Method achieves accuracy comparable to least-square methods near critical points.

It effectively analyzes data unsuitable for polynomial least-squares fitting.

Confirms universality of the 2D Ising model's finite-size scaling function.

Abstract

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.