On Several Problems in Random Matrix Theory and Statistical Mechanics

Download or read book On Several Problems in Random Matrix Theory and Statistical Mechanics written by Yuanyuan Xu and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Random Matrix Theory(RMT) is a fast developing area of modern Mathematics with deep connections to Probability, Statistical Mechanics, Quantum Theory, Number Theory, Statistics, and Integrable Systems. In the first part of my dissertation, I consider an interacting particle system on the unit circle with stronger repulsion than that of the Circular beta Ensemble in RMT and prove the Gaussian approximation of the distribution of the particles. In addition, the Central Limit Theorem(CLT) for the linear statistics of the particles is obtained as a corollary. In the second part of the dissertation, I consider the orthogonal group SO(2n) with the Haar measure and prove the CLT for the linear eigenvalue statistics in the mesoscopic regime where the test function depends on n. The results can be generalized to other classic compact groups, such as SO(2n+1) and Sp(n).