Random matrix

introduction

• The ensembles of random matrices obtained are called Gaussian Orthogonal (GOE), Unitary (GUE), and Symplectic (GSE) Ensembles for = 1, = 2, and = 4 respectively.
• Catalan numbers and random matrices

random self-adjoint matrices

• Wigner matrices
• Band magtrices
• Wishart matrix
• Heavy tails matrices
• Adjacency matrix of Erdos-Renyi graph

Gaussian Wigner matrices

• http://www.math.ucla.edu/~shlyakht/berkeley-07/conference/contrib/peche-talk.pdf
• http://www.math.ucla.edu/~shlyakht/berkeley-07/conference/contrib/guionnet-talk.pdf

Gaussian Unitary Ensemble(GUE) hypothesis

• 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
• GUE Tracy-Widom distribution
  • eigenvalue distributions of the classical Gaussian random matrices ensembles
  • distribution of their largest eigenvalue in the limit of large matrices

\[F_2(s)=\exp\left(-\int_{s}^{\infty}(x-s)q^2(x)dx\right)\] \[F_1(s)=\exp\left(-\frac{1}{2}\int_{s}^{\infty}q(x)dx\right)F_2(s)^{1/2}\] \[F_4(s/\sqrt{2})=\cosh\left(\frac{1}{2}\int_{s}^{\infty}q(x)dx\right)F_2(s)^{1/2}\]

• Painleve II equation

\[q''(s)=sq(s)+2q(s)^3\]

determinantal processes

• Random matrices and determinantal processes http://arxiv.org/abs/math-ph/0510038
• http://terrytao.wordpress.com/2009/08/23/determinantal-processes/

history

• 1920-30 studied by statisticians
• 1950 nuclear physics to describe the energy levels distribution of heavy nuclei
• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• non-intersecting paths
• 3091026
• Macdonald theory
• 리만가설

encyclopedia

• http://mathworld.wolfram.com/WignersSemicircleLaw.html

books

• Large random matrices: lectures on macroscopic asymptotics http://www.mathematik.uni-muenchen.de/~lerdos/SS09/Random/guionnetcours.pdf

expositions

• “At the Far Ends of a New Universal Law | Quanta Magazine.” Accessed October 28, 2014. http://www.quantamagazine.org/20141015-at-the-far-ends-of-a-new-universal-law/.
• 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

articles

• A Note on the Eigenvalue Density of Random MatricesMichael K.-H. Kiessling and Herbert Spohn