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

Abstract

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 $\alpha$ governing the speed at which the sequence approaches criticality is below a certain threshold ${\alpha }_0$. However, when $\alpha$ exceeds ${\alpha }_0$, 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 $0<\alpha <{\alpha }_0$ and via a weak-convergence limit when $\alpha >{\alpha }_0$. To the best of our knowledge, our results are the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model.

Published in at http://dx.doi.org/10.1214/10-AAP679 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model
• Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points
• Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model
• Multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point

Cited by in corpus (6)

• Anomalous mean-field behavior of the fully connected Ising model
• Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling
• Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits
• Refined Asymptotics of the Finite-Size Magnetization via a New Conditional Limit Theorem for the Spin
• Critical fluctuations of noisy period-doubling maps
• Mixing times for the Swapping Algorithm on the Blume-Emery-Griffiths Model