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

Nawaz, Tayyab

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https://hdl.handle.net/2142/97544 Copy

Description

Title

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

Author(s)

Nawaz, Tayyab

Issue Date

2017-04-07

Director of Research (if dissertation) or Advisor (if thesis)

Kirkpatrick, Kay

Doctoral Committee Chair(s)

DeVille, Lee

Committee Member(s)

• Rapti, Zoi
• Hirani, Anil

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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, 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.

Graduation Semester

2017-05

Type of Resource

text

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Copyright and License Information

Copyright 2017 Tayyab Nawaz

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