Random Matrix Theory with an External Source

Nonfiction, Science & Nature, Mathematics, Group Theory, Science, Physics, Mathematical Physics
Cover of the book Random Matrix Theory with an External Source by Edouard Brézin, Shinobu Hikami, Springer Singapore
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Author: Edouard Brézin, Shinobu Hikami ISBN: 9789811033162
Publisher: Springer Singapore Publication: January 11, 2017
Imprint: Springer Language: English
Author: Edouard Brézin, Shinobu Hikami
ISBN: 9789811033162
Publisher: Springer Singapore
Publication: January 11, 2017
Imprint: Springer
Language: English

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.

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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.

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