# Random Matrix Theory with an External Source

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 19)

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Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 19)

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 Gromov–Witten 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.

Random matrix theory Gaussian random matrix models 2D quantum gravity Kontsevich Airy matrix model Gromov-Witten invariants

- DOI https://doi.org/10.1007/978-981-10-3316-2
- Copyright Information The Author(s) 2016
- Publisher Name Springer, Singapore
- eBook Packages Mathematics and Statistics
- Print ISBN 978-981-10-3315-5
- Online ISBN 978-981-10-3316-2
- Series Print ISSN 2197-1757
- Series Online ISSN 2197-1765
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