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
  • \(q''(s)=sq(s)+2q(s)^3\) Painleve II equation
    \(F_2(s)=\exp\left(-\int_{s}^{\infty}(x-s)q^2(x)dx\right)\)
    \(F_1(s)=\exp\left(-\frac{1}{2}\int_{s}^{\infty}q(x)dx\right)F_2(s)^{1/2}\)
    \(F_4(s/\sqrt{2})=\cosh\left(\frac{1}{2}\int_{s}^{\infty}q(x)dx\right)F_2(s)^{1/2}\)
    

 

 

 

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 paths3091026
• Macdonald theory
• 리만가설

 

 

encyclopedia

• http://mathworld.wolfram.com/WignersSemicircleLaw.html
• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

• Large random matrices: lectures on macroscopic asymptotics http://www.mathematik.uni-muenchen.de/~lerdos/SS09/Random/guionnetcours.pdf
• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

expositions

• 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

 

 

articles

• A Note on the Eigenvalue Density of Random MatricesMichael K.-H. Kiessling and Herbert Spohn
  
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/10.1007/s002200050516

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/