ASYMPTOTIC BEHAVIOR OF THE FINITE-SIZE MAGNETIZATION AS A FUNCTION OF THE SPEED OF APPROACH TO CRITICALITY

• 成果类型: Article
• 署名作者: Ellis, Richard S.; Machta, Jonathan; Otto, Peter Tak-Hun
• 署名单位: University of Massachusetts System; University of Massachusetts Amherst; University of Massachusetts System; University of Massachusetts Amherst; Willamette University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP679
• 发表日期: 2010
• 页码: 2118-2161
• 关键词: first-order transitions; LIMIT-THEOREMS; ising systems; triplet ions; phase-transitions; MODEL

摘要

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.