This is a first book to show that the theory of the Gaussian random matrix is essential to understand the universal correlations with random fluctuations and to demonstrate that it is useful to evaluate topological universal quantities. We consider Gaussian random matrix models in the presence of a deterministic matrix source. In such models the correlation functions are known exactly for an arbitrary source and for any size of the matrices. The freedom given by the external source allows for various tunings to different classes of universality. The main interest is to use this freedom to compute various topological invariants for surfaces such as the intersection numbers for curves drawn on a surface of given genus with marked points, Euler characteristics, and the GromovWitten invariants. A remarkable duality for the average of characteristic polynomials is essential for obtaining such topological invariants. The analysis is extended to nonorientable surfaces and to surfaces with boundaries.

Expresses the correlation function of the Gaussian random matrix model with an external source in the integral formula Examines universal behaviors of level spacing distributions for an arbitrary external source Obtains the topological invariants such as the intersection numbers of the moduli space of spin curves by the duality and replica method in the scaling limit of the external source Includes supplementary material: sn.pub/extras