Date of Award

9-2013

Document type

dissertation

Access Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

First Advisor

Richard S. Ellis

Second Advisor

Jonathan Machta

Third Advisor

Luc Rey-Bellet

Subject Categories

Mathematics | Physics

Abstract

In this dissertation four results are presented on the fluctuations of the spin per site around the thermodynamic magnetization in the mean-field Blume-Capel model, a basic model in statistical mechanics. The first two results refine the main theorem in a 2010 paper by R. S. Ellis, J. Machta, and P. T. Otto published in Annals of Applied Probability 20 (2010) 2118-2161. This paper provides the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model. The first main result studies the asymptotics of the centered, finite-size magnetization, giving its precise rate of convergence to 0 along parameter sequences lying in the phase-coexistence region and converging sufficiently slowly to either a second-order point or the tricritical point of the model. A simple inequality yields our second main result, which generalizes the main theorem in the Ellis-Machta-Otto paper by giving an upper bound on the rate of convergence to 0 of the absolute value of the difference between the finite-size magnetization and the thermodynamic magnetization. These first two results have direct relevance to the theory of finite-size scaling. They are consequences of the third main result. This is a new conditional limit theorem for the spin per site, where the conditioning allows us to focus on a neighborhood of the pure states having positive thermodynamic magnetization. The fourth main result is a conditional central limit theorem showing that the fluctuations of the spin per site are Gaussian in a neighborhood of the pure states having positive thermodynamic magnetization.

DOI

https://doi.org/10.7275/cyyd-zq76

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