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

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 $α$ governing the speed at which the sequence approaches criticality is below a certain threshold $α_0$. However, when $α$ exceeds $α_0$, 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 $0<α<α_0$ and via a weak-convergence limit when $α>α_0$. 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.

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