Applications of Random Matrices in Physics

By: Ã?douard Brezin (Editor), Vladimir Kazakov (Editor), Didina Serban (Editor)

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Paperback | 10 March 2006

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524 Pages

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Random matrices are widely and successfully used in physics for almost 60-70 years, beginning with the works of Dyson and Wigner. Although it is an old subject, it is constantly developing into new areas of physics and mathematics. It constitutes now a part of the general culture of a theoretical physicist. Mathematical methods inspired by random matrix theory become more powerful, sophisticated and enjoy rapidly growing applications in physics. Recent examples include the calculation of universal correlations in the mesoscopic system, new applications in disordered and quantum chaotic systems, in combinatorial and growth models, as well as the recent breakthrough, due to the matrix models, in two dimensional gravity and string theory and the non-abelian gauge theories. The book consists of the lectures of the leading specialists and covers rather systematically many of these topics. It can be useful to the specialists in various subjects using random matrices, from PhD students to confirmed scientists.

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You Can Find This Book In

Non-Fiction Mathematics Probability & Statistics Science Chemistry Physics Mathematical Physics Nuclear Physics Materials & States of Matter Engineering & Technology Mechanical Engineering & Materials Materials Science

Preface.- Random Matrices and Number Theory
Introduction
Characteristic polynomials of random unitary matrices
Other compact groups
Families of L-functions and Symmetry
Asymptotic expansions. References
2D Quantum Gravity, Matrix Models and Graph Combinatorics
Introduction
Matrix models for 2D quantum gravity
The one-matrix model I: large N limit and the enumeration of planar graphs
The trees behind the graphs
The one-matrix model II: topological expansions and quantum gravity
The combinatorics beyond matrix models: geodesic distance in planar graphs
Planar graphs as spatial branching processes
Conclusion
Eigenvalue Dynamics, Follytons and Large N Limits of Matrices
References
Random Matrices and Supersymmetry in Disordered Systems. Supersymmetry method
Wave functions fluctuations in a finite volume
Multifractality
Recent and possible future developments
Summary
Acknowledgements
References
Hydrodynamics of Correlated Systems
Introduction
Instanton or rare fluctuation method
Hydrodynamic approach
Linearized hydrodynamics or bosonization
EFP through an asymptotics of the solution
Free fermions
Calogero-Sutherland model
Free fermions on the lattice
Conclusion
Acknowledgements
Appendix: Hydrodynamic approach to non-Galilean invariant systems
Appendix: Exact results for EFP in some integrable models
References
QCD, Chiral Random Matrix Theory and Integrability
Summary
Introduction
QCD
The Dirac Spectrum in QCD
Low Energy Limit of QCD
Chiral RMT and the QCD Dirac Spectrum
Integrability and the QCD Partition Function
QCD at Finite Baryon Density
Full QCD at Nonzero Chemical Potential
Conclusions
Acknowledgements
References
Euclidan Random Matrices: Solved and Open Problems
Introduction
Basic definitions
Physical motivations
Field theory
The simplest case
Phonnos. References
Matrix Models and Growth Processes
Introduction
Some ensembles of random matrices with complex eigenvalues
Exact results at finite N
Large N limit
The matrix model as a growth problem. References
Matrix Models and Topological Strings
Introduction
Matrix models
Type B topological strings and matrix models
Type A topological strings, Chern-Simons theory and matrix models
Matrix Models of Moduli Space
Introduction
Moduli Space of Riemann Surfaces and its Topology
Quadratic Differentials and Fatgraphs
The Penner model
Penner Model and Matrix Gamma Function
The Kontsevich Model
Applications to String Theory
Conclusions. References
Matrix Models and 2D String Theory
Introduction
An overview of string theory
Strings in D-dimensional spacetime
Discretized surfaces and 2D string theory
An overview of observables
Sample calculation: the disk one-point function
Worldsheet description of matrix eigenvalues
Further results
Open problems
References
Matrix Models as Conformal Field Theories
Introduction and historical notes
Hermitian matrix integral: saddle points and hyperellptic curves
The hermitian matrix model as a chiral CFT
Quasiclassical expansions: CFT on a hyperelliptic Riemann surface
Generalization to chains of random matrices
References
Table of Contents provided by Publisher. All Rights Reserved.

Applications of Random Matrices in Physics

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