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 α0. However, when α exceeds α0, 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 < α0 and via a weak-convergence limit when α > α0. 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.
"Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality." Ann. Appl. Probab. 20 (6) 2118 - 2161, December 2010. https://doi.org/10.1214/10-AAP679