"Random matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 4번째 줄: | 4번째 줄: | ||
* Catalan numbers and random matrices | * Catalan numbers and random matrices | ||
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==random self-adjoint matrices== | ==random self-adjoint matrices== | ||
| 16번째 줄: | 16번째 줄: | ||
* Adjacency matrix of Erdos-Renyi graph | * Adjacency matrix of Erdos-Renyi graph | ||
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==Gaussian Wigner matrices== | ==Gaussian Wigner matrices== | ||
| 25번째 줄: | 25번째 줄: | ||
* [http://www.math.ucla.edu/%7Eshlyakht/berkeley-07/conference/contrib/guionnet-talk.pdf http://www.math.ucla.edu/~shlyakht/berkeley-07/conference/contrib/guionnet-talk.pdf] | * [http://www.math.ucla.edu/%7Eshlyakht/berkeley-07/conference/contrib/guionnet-talk.pdf http://www.math.ucla.edu/~shlyakht/berkeley-07/conference/contrib/guionnet-talk.pdf] | ||
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==Gaussian Unitary Ensemble(GUE) hypothesis== | ==Gaussian Unitary Ensemble(GUE) hypothesis== | ||
* Wigner's work on neutron scattering resonances | * Wigner's work on neutron scattering resonances | ||
| − | * Hugh Montgomety and Freeman Dyson | + | * Hugh Montgomety and Freeman Dyson |
** pair correlation function of zeroes of riemann zeta function | ** pair correlation function of zeroes of riemann zeta function | ||
* 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 |
** eigenvalue distributions of the classical Gaussian random matrices ensembles | ** eigenvalue distributions of the classical Gaussian random matrices ensembles | ||
** distribution of their largest eigenvalue in the limit of large matrices | ** distribution of their largest eigenvalue in the limit of large matrices | ||
| − | ** <math>q''(s)=sq(s)+2q(s)^3</math> Painleve II equation | + | ** <math>q''(s)=sq(s)+2q(s)^3</math> Painleve II equation |
| + | :<math>F_2(s)=\exp\left(-\int_{s}^{\infty}(x-s)q^2(x)dx\right)</math> | ||
| + | :<math>F_1(s)=\exp\left(-\frac{1}{2}\int_{s}^{\infty}q(x)dx\right)F_2(s)^{1/2}</math> | ||
| + | :<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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==determinantal processes== | ==determinantal processes== | ||
| 51번째 줄: | 54번째 줄: | ||
* http://terrytao.wordpress.com/2009/08/23/determinantal-processes/ | * http://terrytao.wordpress.com/2009/08/23/determinantal-processes/ | ||
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==history== | ==history== | ||
| 61번째 줄: | 64번째 줄: | ||
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
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==related items== | ==related items== | ||
| − | * [[non-intersecting paths]][[ | + | * [[non-intersecting paths]] |
| + | * [[3091026]] | ||
* [[Macdonald theory]] | * [[Macdonald theory]] | ||
* [http://pythagoras0.springnote.com/pages/4161721 리만가설] | * [http://pythagoras0.springnote.com/pages/4161721 리만가설] | ||
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==encyclopedia== | ==encyclopedia== | ||
| 79번째 줄: | 83번째 줄: | ||
* http://mathworld.wolfram.com/WignersSemicircleLaw.html | * http://mathworld.wolfram.com/WignersSemicircleLaw.html | ||
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==books== | ==books== | ||
| 86번째 줄: | 90번째 줄: | ||
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==expositions== | ==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 | * Random matrices as a paradigm | ||
* http://www.phys.ust.hk/yilong/research/PhaseSpaceNetHan.pdf | * http://www.phys.ust.hk/yilong/research/PhaseSpaceNetHan.pdf | ||
| 95번째 줄: | 99번째 줄: | ||
* [http://math.arizona.edu/events/AZschool/material/AZ10-erdos.pdf Universality of Wigner Random Matrices: a Survey of Recent Results] | * [http://math.arizona.edu/events/AZschool/material/AZ10-erdos.pdf Universality of Wigner Random Matrices: a Survey of Recent Results] | ||
* http://www.mathematik.uni-muenchen.de/~lerdos/SS09/Random/plan.html | * http://www.mathematik.uni-muenchen.de/~lerdos/SS09/Random/plan.html | ||
| − | * Introduction to Random Matrix | + | * 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== | ||
* [http://dx.doi.org/10.1007/s002200050516%20 A Note on the Eigenvalue Density of Random Matrices]Michael K.-H. Kiessling and Herbert Spohn | * [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]] | ||
2014년 10월 28일 (화) 04:49 판
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=
encyclopedia
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
- A Note on the Eigenvalue Density of Random MatricesMichael K.-H. Kiessling and Herbert Spohn