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The main focus of this paper is to determine whether the thermodynamicmagnetization is a physically relevant estimator of the finite-sizemagnetization. This is done by comparing the asymptotic behaviors of these twoquantities along parameter sequences converging to either a second-order pointor the tricritical point in the mean-field Blume--Capel model. We show that thethermodynamic magnetization and the finite-size magnetization are asymptoticwhen the parameter $\alpha$ governing the speed at which the sequenceapproaches criticality is below a certain threshold $\alpha_0$. However, when$\alpha$ exceeds $\alpha_0$, the thermodynamic magnetization converges to 0much faster than the finite-size magnetization. The asymptotic behavior of thefinite-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 confirmationof the statistical mechanical theory of finite-size scaling for a mean-fieldmodel.

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