Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality

TL;DR

This paper rigorously analyzes how finite-size magnetization behaves near critical points in the mean-field Blume--Capel model, revealing the conditions under which it aligns with thermodynamic magnetization and confirming finite-size scaling theory.

Contribution

It provides the first rigorous proof of finite-size scaling behavior for a mean-field model, relating the asymptotics of finite-size and thermodynamic magnetizations.

Findings

Finite-size and thermodynamic magnetizations are asymptotic when approaching criticality slowly.

Thermodynamic magnetization converges faster to zero than finite-size magnetization when approaching quickly.

Asymptotic behavior is characterized using moderate deviation principles and weak convergence.

Abstract

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume--Capel model. We show that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the parameter $\alpha$ governing the speed at which the sequence approaches criticality is below a certain threshold $\alpha_0$. However, when $\alpha$ exceeds $\alpha_0$, the thermodynamic magnetization converges to 0 much faster than the finite-size magnetization. The asymptotic behavior of the finite-size magnetization is proved via a moderate deviation principle when $0<\alpha<\alpha_0$ and via a weak-convergence limit when $\alpha >\alpha_0$. To the best of our knowledge, our results are the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model.