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

• Tracy-Widom distribution

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
• Random matrix theory over finite fields

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

• 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

articles

• 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.