Random Matrix Theory (RMT) is a mathematical framework used to study the statistical properties of eigenvalues of large matrices, particularly in physics. It has applications in various fields, including nuclear physics, quantum mechanics, and disordered systems. The theory considers two main classes of random matrices: Hermitian and non-Hermitian. Hermitian matrices have real eigenvalues and are characterized by the Dyson index β = 1, 2, or 4, corresponding to different symmetry classes. Non-Hermitian matrices, on the other hand, have eigenvalues distributed in the complex plane and are used to model systems with complex interactions. The Wigner-Dyson ensembles are the most well-known Hermitian random matrix ensembles, characterized by the probability distribution P(H)DH = N e^{-Nβ/4 Tr H†H} DH. These ensembles include the Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), and Gaussian Symplectic Ensemble (GSE). The GOE is also known as the Wishart distribution. The chiral ensembles, which are relevant in quantum chromodynamics (QCD), have a block structure and are defined by a specific probability distribution involving the determinant of a matrix. The Altland-Zirnbauer ensembles describe disordered superconductors and have a block structure similar to the chiral ensembles. Mathematical methods for analyzing random matrices include orthogonal polynomials, Selberg integrals, and the supersymmetric method. The supersymmetric method is particularly useful for calculating spectral correlation functions and has been applied to problems where the orthogonal polynomial method fails. The replica trick is another method used to study eigenvalue correlations, although it has limitations in certain cases. Resolvent expansion methods and Hermitization are also employed to analyze the properties of random matrices. Applications of RMT include nuclear level spacings, where the statistical fluctuations of energy levels are found to be universal and described by RMT predictions. The theory has also been applied to the hydrogen atom in a magnetic field, where the spectral correlations transition from Poisson to RMT behavior as the magnetic field increases. Additionally, RMT is used to study quantum chaos in systems such as billiards, where the classical dynamics are chaotic and the quantum mechanical properties exhibit universal correlations. The theory provides a powerful framework for understanding the statistical properties of complex systems and has been widely applied in various areas of physics.Random Matrix Theory (RMT) is a mathematical framework used to study the statistical properties of eigenvalues of large matrices, particularly in physics. It has applications in various fields, including nuclear physics, quantum mechanics, and disordered systems. The theory considers two main classes of random matrices: Hermitian and non-Hermitian. Hermitian matrices have real eigenvalues and are characterized by the Dyson index β = 1, 2, or 4, corresponding to different symmetry classes. Non-Hermitian matrices, on the other hand, have eigenvalues distributed in the complex plane and are used to model systems with complex interactions. The Wigner-Dyson ensembles are the most well-known Hermitian random matrix ensembles, characterized by the probability distribution P(H)DH = N e^{-Nβ/4 Tr H†H} DH. These ensembles include the Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), and Gaussian Symplectic Ensemble (GSE). The GOE is also known as the Wishart distribution. The chiral ensembles, which are relevant in quantum chromodynamics (QCD), have a block structure and are defined by a specific probability distribution involving the determinant of a matrix. The Altland-Zirnbauer ensembles describe disordered superconductors and have a block structure similar to the chiral ensembles. Mathematical methods for analyzing random matrices include orthogonal polynomials, Selberg integrals, and the supersymmetric method. The supersymmetric method is particularly useful for calculating spectral correlation functions and has been applied to problems where the orthogonal polynomial method fails. The replica trick is another method used to study eigenvalue correlations, although it has limitations in certain cases. Resolvent expansion methods and Hermitization are also employed to analyze the properties of random matrices. Applications of RMT include nuclear level spacings, where the statistical fluctuations of energy levels are found to be universal and described by RMT predictions. The theory has also been applied to the hydrogen atom in a magnetic field, where the spectral correlations transition from Poisson to RMT behavior as the magnetic field increases. Additionally, RMT is used to study quantum chaos in systems such as billiards, where the classical dynamics are chaotic and the quantum mechanical properties exhibit universal correlations. The theory provides a powerful framework for understanding the statistical properties of complex systems and has been widely applied in various areas of physics.