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Title:Applications of Stein's method and large deviations principle's in mean-field O(N) models
Author(s):Nawaz, Tayyab
Director of Research:Kirkpatrick, Kay
Doctoral Committee Chair(s):DeVille, Lee
Doctoral Committee Member(s):Rapti, Zoi; Hirani, Anil
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mean-field
Rate function
Total spin
Limit theorem
Phase transition
Abstract:In the first part of this thesis, we will discuss the classical XY model on complete graph in the mean-field (infinite-vertex) limit. Using theory of large deviations and Stein's method, in particular, Cram\'er and Sanov-type results, we present a number of results coming from the limit theorems with rates of convergence, and phase transition behavior for classical XY model. In the second part, we will generalize our results to mean-field classical $N$-vector models, for integers $N \ge 2$. We will use the theory of large deviations and Stein's method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Some of the important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in \cite{KM} but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.
Issue Date:2017-04-07
Type:Thesis
URI:http://hdl.handle.net/2142/97544
Rights Information:Copyright 2017 Tayyab Nawaz
Date Available in IDEALS:2017-08-10
Date Deposited:2017-05


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