Applications of Stein's method and large deviations principle's in mean-field O(N) models

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, Cramér 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 [KM13] but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Nawaz, Tayyab

Contributors dc:contributor

• Kirkpatrick, Kay
• DeVille, Lee
• Rapti, Zoi
• Hirani, Anil

• Mean-field
• Rate function
• Total spin
• Limit theorem
• Phase transition

Statement dc:rights

• Copyright 2017 Tayyab Nawaz

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/97544

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/97544