"Random matrix"의 두 판 사이의 차이

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==introduction==
 
==introduction==
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* 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.
 
* The ensembles of random matrices obtained are called Gaussian Orthogonal (GOE), Unitary (GUE), and Symplectic (GSE) Ensembles for = 1, = 2, and = 4 respectively.
 
* The ensembles of random matrices obtained are called Gaussian Orthogonal (GOE), Unitary (GUE), and Symplectic (GSE) Ensembles for = 1, = 2, and = 4 respectively.
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* Catalan numbers and random matrices
 
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<blockquote>
 
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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.
 
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</blockquote>
 
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* GUE is a big open problem but proven for random matrix models
 
* GUE is a big open problem but proven for random matrix models
 
==GUE Tracy-Widom distribution==
 
==GUE Tracy-Widom distribution==
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* eigenvalue distributions of the classical Gaussian random matrices ensembles
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* [[Tracy-Widom distribution]]
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* distribution of their largest eigenvalue in the limit of large matrices
 
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:<math>F_2(s)=\exp\left(-\int_{s}^{\infty}(x-s)q^2(x)dx\right)</math>
 
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:<math>F_1(s)=\exp\left(-\frac{1}{2}\int_{s}^{\infty}q(x)dx\right)F_2(s)^{1/2}</math>
 
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:<math>F_4(s/\sqrt{2})=\cosh\left(\frac{1}{2}\int_{s}^{\infty}q(x)dx\right)F_2(s)^{1/2}</math>
 
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*  Painleve II equation
 
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:<math>q''(s)=sq(s)+2q(s)^3</math>
 
  
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<blockquote>
 
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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.
 
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</blockquote>
 
  
 
==determinantal processes==
 
==determinantal processes==
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==memo==
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* Catalan numbers and random matrices
  
 
   
 
   
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* 1920-30 studied by statisticians
 
* 1920-30 studied by statisticians
 
* 1950 nuclear physics to describe the energy levels distribution of heavy nuclei
 
* 1950 nuclear physics to describe the energy levels distribution of heavy nuclei
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
  
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==related items==
 
==related items==
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* [[Macdonald theory]]
 
* [[Macdonald theory]]
 
* [http://pythagoras0.springnote.com/pages/4161721 리만가설]
 
* [http://pythagoras0.springnote.com/pages/4161721 리만가설]
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* [[Random matrix theory over finite fields]]
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* [[Gaussian Orthogonal Ensemble]]
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==encyclopedia==
 
==encyclopedia==
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==expositions==
 
==expositions==
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* “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/.
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* Gernot Akemann, Random Matrix Theory and Quantum Chromodynamics, http://arxiv.org/abs/1603.06011v1
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* Florent Benaych-Georges, Antti Knowles, Lectures on the local semicircle law for Wigner matrices, http://arxiv.org/abs/1601.04055v2
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* http://arxiv.org/abs/1601.03678
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* 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.
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* 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.
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* 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
 
* Random matrices as a paradigm
 
* http://www.phys.ust.hk/yilong/research/PhaseSpaceNetHan.pdf
 
* http://www.phys.ust.hk/yilong/research/PhaseSpaceNetHan.pdf
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* 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
 
* 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
 
* http://stuff.mit.edu/people/raj/Acta05rmt.pdf
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==articles==
 
==articles==
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* Miklós Kornyik, György Michaletzky, Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law, arXiv:1512.03724 [math.CA], November 23 2015, http://arxiv.org/abs/1512.03724
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* Percy Deift, Thomas Trogdon, Universality for the Toda algorithm to compute the eigenvalues of a random matrix, arXiv:1604.07384 [math.PR], April 25 2016, http://arxiv.org/abs/1604.07384
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* 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.
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* 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.
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* 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.
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* [http://dx.doi.org/10.1007/s002200050516%20 A Note on the Eigenvalue Density of Random Matrices], Michael K.-H. Kiessling and Herbert Spohn
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* 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.
  
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* [http://dx.doi.org/10.1007/s002200050516%20 A Note on the Eigenvalue Density of Random Matrices]Michael K.-H. Kiessling and Herbert Spohn
 
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[[분류:개인노트]]
 
[[분류:개인노트]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q176604 Q176604]
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===Spacy 패턴 목록===
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* [{'LOWER': 'random'}, {'LEMMA': 'matrix'}]

2021년 2월 17일 (수) 03:10 기준 최신판

introduction

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


random self-adjoint matrices

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



Gaussian Wigner matrices



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


determinantal processes


memo

  • Catalan numbers and random matrices


history

  • 1920-30 studied by statisticians
  • 1950 nuclear physics to describe the energy levels distribution of heavy nuclei


related items

encyclopedia


books



expositions

articles

  • Miklós Kornyik, György Michaletzky, Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law, arXiv:1512.03724 [math.CA], November 23 2015, http://arxiv.org/abs/1512.03724
  • Percy Deift, Thomas Trogdon, Universality for the Toda algorithm to compute the eigenvalues of a random matrix, arXiv:1604.07384 [math.PR], April 25 2016, http://arxiv.org/abs/1604.07384
  • 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.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'random'}, {'LEMMA': 'matrix'}]
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