Random matrix

introduction

Random Matrix Theory is a paradigm for describing and understanding a variety of phenomena in physics, mathematics, and potentially other disciplines. The theory was born in the early 1950s when theoretical physicist Eugene Wigner suggested that the problem of determining the energy level spacings of heavy nuclei - intractable by analytic means - might be modeled after the spectrum of a large random matrix.
The ensembles of random matrices obtained are called Gaussian Orthogonal (GOE), Unitary (GUE), and Symplectic (GSE) Ensembles for = 1, = 2, and = 4 respectively.

Wigner's work on neutron scattering resonances
Hugh Montgomety and Freeman Dyson
- pair correlation function of zeroes of riemann zeta function
GUE is a big open problem but proven for random matrix models

Gernot Akemann, Random Matrix Theory and Quantum Chromodynamics, http://arxiv.org/abs/1603.06011v1
Florent Benaych-Georges, Antti Knowles, Lectures on the local semicircle law for Wigner matrices, http://arxiv.org/abs/1601.04055v2
http://arxiv.org/abs/1601.03678
Diaconis, Persi, and Peter J. Forrester. “A. Hurwitz and the Origins of Random Matrix Theory in Mathematics.” arXiv:1512.09229 [math-Ph], December 31, 2015. http://arxiv.org/abs/1512.09229.
Zyczkowski, K., and M. Kus. “Random Unitary Matrices.” Journal of Physics A: Mathematical and General 27, no. 12 (1994): 4235. doi:10.1088/0305-4470/27/12/028.
Mezzadri, Francesco. “How to Generate Random Matrices from the Classical Compact Groups.” arXiv:math-ph/0609050, September 18, 2006. http://arxiv.org/abs/math-ph/0609050.
Random matrices as a paradigm
http://www.phys.ust.hk/yilong/research/PhaseSpaceNetHan.pdf
http://www.ims.nus.edu.sg/Programs/randommatrix/files/sverdu_p.pdf
Universality of Wigner Random Matrices: a Survey of Recent Results
http://www.mathematik.uni-muenchen.de/~lerdos/SS09/Random/plan.html
Introduction to Random Matrix Theory from An Invitation to Modern Number Theory http://web.williams.edu/go/math/sjmiller/public_html/BrownClasses/54/handouts/IntroRMT_Math54.pdf
http://stuff.mit.edu/people/raj/Acta05rmt.pdf

Miklós Kornyik, György Michaletzky, Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law, arXiv:1512.03724 [math.CA], November 23 2015, http://arxiv.org/abs/1512.03724
Percy Deift, Thomas Trogdon, Universality for the Toda algorithm to compute the eigenvalues of a random matrix, arXiv:1604.07384 [math.PR], April 25 2016, http://arxiv.org/abs/1604.07384
Forrester, Peter J. “Analogies between Random Matrix Ensembles and the One-Component Plasma in Two-Dimensions.” arXiv:1511.02946 [cond-Mat, Physics:math-Ph], November 9, 2015. http://arxiv.org/abs/1511.02946.
Kakei, Saburo. “Hirota Bilinear Approach to GUE, NLS, and Painlev’e IV.” arXiv:1510.07560 [math-Ph, Physics:nlin], October 26, 2015. http://arxiv.org/abs/1510.07560.
Joyner, Christopher H., and Uzy Smilansky. ‘Dyson’s Brownian-Motion Model for Random Matrix Theory - Revisited. With an Appendix by Don Zagier’. arXiv:1503.06417 [cond-Mat, Physics:math-Ph], 22 March 2015. http://arxiv.org/abs/1503.06417.
A Note on the Eigenvalue Density of Random Matrices, Michael K.-H. Kiessling and Herbert Spohn
Farmer, David W. “On the Neighbor Spacing of Eigenvalues of Unitary Matrices.” arXiv:0709.4529 [math-Ph], September 28, 2007. http://arxiv.org/abs/0709.4529.