RANDOM MATRICES

Abstract

The study of Random Matrix Theory (RMT) has emerged as a cornerstone of modern statistical mechanics, high-dimensional data analysis, and quantum physics. This thesis provides a rigorous examination of the Wigner Semicircle Law, a fundamental result that describes the global distribution of eigenvalues for large symmetric random matrices. The primary objective of this research is to investigate the convergence of the Empirical Spectral Distribution (ESD) toward the theoretical semicircular density as the matrix dimension N approaches infinity. The dissertation begins with a systematic exploration of Gaussian Ensembles, specifically the Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE), establishing the probabilistic framework for eigenvalue interactions. Using the Method of Moments and combinatorial techniques involving Catalan Numbers and non-crossing partitions, we present a detailed outline of the proof of the Semicircle Law. This theoretical foundation is further validated through extensive numerical simulations and computational modeling, demonstrating the robustness of the law across different underlying distributions of matrix entries. Furthermore, the work bridges the gap between abstract probability and practical application by analyzing the law's relevance in Wireless Communication, Financial Correlation Matrices, and Deep Learning stability. The findings underscore the principle of Universality, proving that the semicircular shape remains invariant regardless of the specific distribution of individual entries, provided they are independent and identically distributed (i.i.d.). This research concludes by addressing the limitations of finite-sized systems and proposing future extensions into non-Hermitian matrices and free probability theory.