Refined Asymptotics of the Finite-Size Magnetization via a New Conditional Limit Theorem for the Spin

Richard S. Ellis; Jingran Li

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arXiv · 1205.0970

Refined Asymptotics of the Finite-Size Magnetization via a New Conditional Limit Theorem for the Spin

Richard S. Ellis·Jingran Li

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Abstract

We study the fluctuations of the spin per site around the thermodynamic magnetization in the mean-field Blume-Capel model. Our main theorem generalizes the main result in a previous paper (Ellis, Machta, and Otto) in which the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model is given. In that paper our goal 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. The main result is 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 $α_0$. Our main theorem in the present paper on the fluctuations of the spin per site around the thermodynamic magnetization is based on a new conditional limit theorem for the spin, which is closely related to a new conditional central limit theorem for the spin.

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Richard S. Ellis, Jingran Li. 2012-05-04. Refined Asymptotics of the Finite-Size Magnetization via a New Conditional Limit Theorem for the Spin. https://arxiv.org/abs/1205.0970

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