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=

 

 

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