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

Ellis, Richard S. ; Machta, Jonathan ; Otto, Peter Tak-Hun

Ann. Appl. Probab., Tome 20 (2010) no. 1, p. 2118-2161 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2010-12-15
Classification:  Finite-size magnetization,  thermodynamic magnetization,  second-order phase transition,  first-order phase transition,  tricritical point,  moderate deviation principle,  large deviation principle,  scaling limit,  Blume–Capel model,  finite-size scaling,  60F10,  60F05,  82B20