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
Recommended Citation
Li, Jingran, "Conditional Gaussian Fluctuations and Refined Asymptotics of the Spin In the Phase-Coexistence Region" (2013). Open Access Dissertations. 806.
https://doi.org/10.7275/cyyd-zq76
https://scholarworks.umass.edu/open_access_dissertations/806