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

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

2009-08-07

arxiv.org/abs/0908.1103v2

Annals of Applied Probability 2010, Vol. 20, No. 6, 2118-2161

Mathematics

Probability

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

Scientific paper

10.1214/10-AAP679

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.

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