Integrable Systems, Random Matrices, and Random Processes

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


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.


Integrable System Vertex Operator Random Matrix Random Matrix Theory Fredholm Determinant 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA

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