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 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.

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

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

Harold Widom and Craig Tracy have made basic contributions to the theory. There are three basic classes of Random Matrix models, known as "GUE, GOE, and GSE", for Gaussian Unitary, Orthogonal, and Symplectic ensembles, according to the type of matrix which is to be randomly selected. The limiting distribution (as the size of the matrix tends to infinity) for the largest eigenvaule in the GUE model had been computed via a Fredholm determinant by physicists. Widom and Tracy obtained the limiting distributions in the GOE and GSE models in terms of Fredholm determinants. They also showed that all three limiting distributions were representable in terms of Painleve functions, hence establishing connections to the theory of integrable dynamical systems. The resulting distributions are now universally known as the Tracy-Widom distributions. It is believed to describe new universal limit laws for a wide variety of processes arising in mathematical physics and interacting particle systems. The distribution is destined to play an increasingly important role, akin to the bell curve, or normal distribution so familiar in statistics and probability.

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

• 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 MatricesMichael K.-H. Kiessling and Herbert Spohn