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

TL;DR

This paper extends the understanding of finite-size effects in the mean-field Blume-Capel model by establishing a new conditional limit theorem for spin fluctuations, confirming the relevance of thermodynamic magnetization as an estimator.

Contribution

It introduces a novel conditional limit theorem for the spin, generalizing previous results and advancing the statistical mechanical theory of finite-size scaling in mean-field models.

Findings

Thermodynamic and finite-size magnetizations are asymptotic below a critical threshold.

A new conditional limit theorem for the spin is established.

The results confirm the physical relevance of thermodynamic magnetization as an estimator.

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 $\alpha$ governing the speed at which the sequence approaches criticality is below a certain threshold $\alpha_0$. 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.