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

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

Nawaz, Tayyab

关键词: Mean-field;    Rate function;    Total spin;    Limit theorem;    Phase transition;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/97544/NAWAZ-DISSERTATION-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.