The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 20, Number 6 (2010), 2118-2161.
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 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.
Ann. Appl. Probab. Volume 20, Number 6 (2010), 2118-2161.
First available in Project Euclid: 19 October 2010
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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
Ellis, Richard S.; Machta, Jonathan; Otto, Peter Tak-Hun. Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality. Ann. Appl. Probab. 20 (2010), no. 6, 2118--2161. doi:10.1214/10-AAP679. https://projecteuclid.org/euclid.aoap/1287494556