Date of Award

9-2013

Document 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.

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