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Integrable Systems, Random Matrices, and Random Processes

  • Mark AdlerEmail author
Chapter
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Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

Random matrix theory began in the 1950s, when E. Wigner [58] proposed that the local statistical behavior of scattering resonance levels for neutrons off heavy nuclei could be modeled by the statistical behavior of eigenvalues of a large random matrix, provided the ensemble had orthogonal, unitary or symplectic invariance. The approach was developed by many others, like Dyson [30, 31], Gaudin [34] and Mehta, as documented in Mehta’s [44] fa-mous treatise. The field experienced a revival in the 1980s due to the work of M. Jimbo, T. Miwa, Y. Mori, and M. Sato [36, 37], showing the Fredholm determinant involving the sine kernel, which had appeared in random ma-trix theory for large matrices, satisfied the fifth Painleve transcendent; thus linking random matrix theory to integrable mathematics. Tracy and Widom soon applied their ideas, using more efficient function-theoretic methods, to the largest eigenvalues of unitary, orthogonal and symplectic matrices in the limit of large matrices, with suitable rescaling. This lead to the TracyWidom distributions for the 3 cases and the modern revival of random matrix theory (RMT) was off and running.

Keywords

Integrable System Vertex Operator Random Matrix Random Matrix Theory Fredholm Determinant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA

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