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随机矩阵

二月 06, 2023

提示：此条目的主题不是轉移矩陣。

在概率論和數學物理中，隨機矩陣（英語：Random matrix）是一個矩陣值的随机变量，也就是说，一个矩阵中的所有元素都是随机变量。[1]

應用

物理

其他（AI、数学、统计）

随机矩阵模特

设 是 的矩阵，有下面的概率测度：

 

例子，高斯模特： 。

參見

阅读

參考文獻

  1. ^ 1.0 1.1 Terence Tao 陶哲轩. (PDF). （原始内容 (PDF)存档于2021-05-06） （英语）. 
  2. ^ Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079. 
  3. ^ Mehta, M.L. Random Matrices. Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7. 
  4. ^ Bohigas, O.; Giannoni, M.J.; Schmit, Schmit. Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws. Phys. Rev. Lett. 1984, 52 (1): 1–4. Bibcode:1984PhRvL..52....1B. doi:10.1103/PhysRevLett.52.1. 
  5. ^ Aaronson, Scott; Arkhipov, Alex. The computational complexity of linear optics. Theory of Computing. 2013, 9: 143–252. doi:10.4086/toc.2013.v009a004. 
  6. ^ Russell, Nicholas; Chakhmakhchyan, Levon; O'Brien, Jeremy; Laing, Anthony. Direct dialling of Haar random unitary matrices. New J. Phys. 2017, 19 (3): 033007. Bibcode:2017NJPh...19c3007R. arXiv:1506.06220  . doi:10.1088/1367-2630/aa60ed. 
  7. ^ Random Matrix Theory and Chiral Symmetry in QCD. Annu. Rev. Nucl. Part. Sci. 2000, 50: 343–410. Bibcode:2000ARNPS..50..343V. arXiv:hep-ph/0003017  . doi:10.1146/annurev.nucl.50.1.343. 
  8. ^ Horizon in random matrix theory, the Hawking radiation, and flow of cold atoms. Phys. Rev. Lett. October 2009, 103 (16): 166401. Bibcode:2009PhRvL.103p6401F. PMID 19905710. arXiv:0905.3533  . doi:10.1103/PhysRevLett.103.166401. 
  9. ^ Magnetic-field asymmetry of nonlinear mesoscopic transport. Phys. Rev. Lett. September 2004, 93 (10): 106802. Bibcode:2004PhRvL..93j6802S. PMID 15447435. arXiv:cond-mat/0404387  . doi:10.1103/PhysRevLett.93.106802. 
  10. ^ Spin torque and waviness in magnetic multilayers: a bridge between Valet-Fert theory and quantum approaches. Phys. Rev. Lett. August 2009, 103 (6): 066602. Bibcode:2009PhRvL.103f6602R. PMID 19792592. arXiv:0902.4360  . doi:10.1103/PhysRevLett.103.066602. 
  11. ^ Callaway DJE. Random matrices, fractional statistics, and the quantum Hall effect. Phys. Rev. B. April 1991, 43 (10): 8641–8643. Bibcode:1991PhRvB..43.8641C. PMID 9996505. doi:10.1103/PhysRevB.43.8641. 
  12. ^ Correlated random band matrices: localization-delocalization transitions. Phys. Rev. E. June 2000, 61 (6 Pt A): 6278–86. Bibcode:2000PhRvE..61.6278J. PMID 11088301. arXiv:cond-mat/9911467  . doi:10.1103/PhysRevE.61.6278. 
  13. ^ Spin-orbit coupling, antilocalization, and parallel magnetic fields in quantum dots. Phys. Rev. Lett. December 2002, 89 (27): 276803. Bibcode:2002PhRvL..89A6803Z. PMID 12513231. arXiv:cond-mat/0208436  . doi:10.1103/PhysRevLett.89.276803. 
  14. ^ Bahcall SR. Random Matrix Model for Superconductors in a Magnetic Field. Phys. Rev. Lett. December 1996, 77 (26): 5276–5279. Bibcode:1996PhRvL..77.5276B. PMID 10062760. arXiv:cond-mat/9611136  . doi:10.1103/PhysRevLett.77.5276. 
  15. ^ Keating, Jon. The Riemann zeta-function and quantum chaology. Proc. Internat. School of Phys. Enrico Fermi. 1993, CXIX: 145–185. ISBN 9780444815880. doi:10.1016/b978-0-444-81588-0.50008-0. 
  16. ^ Wishart, J. Generalized product moment distribution in samples. Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32. 
  17. ^ Tropp, J. User-Friendly Tail Bounds for Sums of Random Matrices. Foundations of Computational Mathematics. 2011, 12 (4): 389–434. arXiv:1004.4389  . doi:10.1007/s10208-011-9099-z. 
  18. ^ von Neumann, J.; Goldstine, H.H. Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6. 
  19. ^ Edelman, A.; Rao, N.R. Random matrix theory. Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236. 
  20. ^ Chow, Gregory P. Analysis and Control of Dynamic Economic Systems. New York: Wiley. 1976. ISBN 0-471-15616-7. 
  21. ^ Turnovsky, Stephen. Optimal stabilization policies for stochastic linear systems: The case of correlated multiplicative and additive disturbances. Review of Economic Studies. 1976, 43 (1): 191–194. JSTOR 2296741. doi:10.2307/2296614. 
  22. ^ 22.0 22.1 Turnovsky, Stephen. The stability properties of optimal economic policies. American Economic Review. 1974, 64 (1): 136–148. JSTOR 1814888. 
  23. ^ García del Molino, Luis Carlos; Pakdaman, Khashayar; Touboul, Jonathan; Wainrib, Gilles. Synchronization in random balanced networks. Physical Review E. October 2013, 88 (4): 042824. Bibcode:2013PhRvE..88d2824G. arXiv:1306.2576  . doi:10.1103/PhysRevE.88.042824. 
  24. ^ Rajan, Kanaka; Abbott, L. Eigenvalue Spectra of Random Matrices for Neural Networks. Physical Review Letters. November 2006, 97 (18): 188104. Bibcode:2006PhRvL..97r8104R. PMID 17155583. doi:10.1103/PhysRevLett.97.188104. 
  25. ^ Wainrib, Gilles; Touboul, Jonathan. Topological and Dynamical Complexity of Random Neural Networks. Physical Review Letters. March 2013, 110 (11): 118101. Bibcode:2013PhRvL.110k8101W. PMID 25166580. arXiv:1210.5082  . doi:10.1103/PhysRevLett.110.118101. 
  26. ^ Muir, Dylan; Mrsic-Flogel, Thomas. (PDF). Phys. Rev. E. 2015, 91 (4): 042808 [2020-01-13]. Bibcode:2015PhRvE..91d2808M. PMID 25974548. doi:10.1103/PhysRevE.91.042808. （原始内容 (PDF)存档于2018-07-21）. 
  27. ^ Cosme Louart, Zhenyu Liao, and Romain Couillet. (PDF). （原始内容 (PDF)存档于2020-01-13）. 
  28. ^ Zhenyu Liao, Romain Couillet. (PDF). （原始内容 (PDF)存档于2020-11-12）. 
  29. ^ Jeffrey Pennington, Pratik Worah. (PDF). （原始内容 (PDF)存档于2020-11-03）. 
  30. ^ Mehta, M.L. Random Matrices. Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7. 
  31. ^ Anderson, G.W.; Guionnet, A.; Zeitouni, O. An introduction to random matrices.. Cambridge: Cambridge University Press. 2010. ISBN 978-0-521-19452-5. 
  32. ^ Akemann, G.; Baik, J.; Di Francesco, P. The Oxford Handbook of Random Matrix Theory.. Oxford: Oxford University Press. 2011. ISBN 978-0-19-957400-1. 
  33. ^ Edelman, A.; Rao, N.R. Random matrix theory. Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236. 
  34. ^ Pastur, L.A. Spectra of random self-adjoint operators. Russ. Math. Surv. 1973, 28 (1): 1–67. Bibcode:1973RuMaS..28....1P. doi:10.1070/RM1973v028n01ABEH001396. 
  35. ^ Diaconis, Persi. Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. American Mathematical Society. Bulletin. New Series. 2003, 40 (2): 155–178. MR 1962294. doi:10.1090/S0273-0979-03-00975-3. 
  36. ^ Diaconis, Persi. . Notices of the American Mathematical Society. 2005, 52 (11): 1348–1349 [2020-01-13]. ISSN 0002-9920. MR 2183871. （原始内容存档于2019-03-28）. 
  37. ^ Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079. 
  38. ^ Wishart, J. Generalized product moment distribution in samples. Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32. 
  39. ^ von Neumann, J.; Goldstine, H.H. Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6. 

二月 06, 2023
随机矩阵, 提示, 此条目的主题不是轉移矩陣, 在概率論和數學物理中, 隨機矩陣, 英語, random, matrix, 是一個矩陣值的随机变量, 也就是说, 一个矩阵中的所有元素都是随机变量, 目录, 應用, 物理, 其他, 数学, 统计, 模特, 參見, 阅读, 參考文獻應用, 编辑物理, 编辑, 原子核物理学, 量子場論, 量子混沌, quantum, chaos, bohigas, giannoni, schmit, 猜想, 量子光學, 米爾斯理論, 量子色動力學, 兩維的量子引力, cft对偶, 介观物. 提示 此条目的主题不是轉移矩陣 在概率論和數學物理中 隨機矩陣 英語 Random matrix 是一個矩陣值的随机变量 也就是说 一个矩阵中的所有元素都是随机变量 1 目录 1 應用 1 1 物理 1 2 其他 AI 数学 统计 2 随机矩阵模特 3 參見 4 阅读 5 參考文獻應用 编辑物理 编辑 原子核物理学 2 3 量子場論 量子混沌 quantum chaos Bohigas Giannoni Schmit BGS 猜想 4 量子光學 5 6 楊 米爾斯理論 量子色動力學 7 兩維的量子引力 AdS CFT对偶 8 介观物理学 9 自旋转移矩 10 小数量子霍爾效果 11 安德森的本地化 Anderson localization 12 量子点 13 超导現象 14 其他 AI 数学 统计 编辑 数论 黎曼z函數和其他L函数的零分布 希尔伯特 波利亚猜想 黎曼猜想 15 多元变量统计 16 17 數值分析 18 19 最优控制 20 21 22 神经科学理论 混沌理论 22 23 24 25 26 人工智能 机器学习 深度学习 深度神经网络 27 28 29 随机矩阵模特 编辑设H displaystyle H 是N N displaystyle N times N 的矩阵 有下面的概率测度 P H 1 Z e N t r V H displaystyle P H frac 1 Z e Ntr V H 例子 高斯模特 V H H 2 2 displaystyle V H H 2 2 GUE Gaussian Unitary Ensemble H是埃尔米特矩阵 通过1 N展开 維格納半圓分布描述H的大N特征值的機率密度函數r E displaystyle rho E 1 GOE Orthogonal H是对称矩阵 GSE Symplectic H是四元数的矩阵 Quaternion matrix 參見 编辑維格納半圓分布 弗里曼 戴森气体模特 Dyson gas model 1 N展开 普遍性 物理学 Universality Spectral Theory 非古典机率 Free probability 阅读 编辑陶哲轩的Topics in random matrix theory https terrytao files wordpress com 2011 02 matrix book pdf 页面存档备份 存于互联网档案馆 其他书 30 31 32 文章 33 34 35 36 原始文章 37 38 39 Voiculescu Free Probability Theory and Operator Algebras Speicher Free Probability Theory https arxiv org pdf 0911 0087 pdf 页面存档备份 存于互联网档案馆 徐一鴻的https en wikipedia org wiki Quantum Field Theory in a Nutshell 页面存档备份 存于互联网档案馆 Large N expansion 參考文獻 编辑 1 0 1 1 Terence Tao 陶哲轩 Topics in random matrix theory PDF 原始内容 PDF 存档于2021 05 06 英语 Wigner E Characteristic vectors of bordered matrices with infinite dimensions Annals of Mathematics 1955 62 3 548 564 JSTOR 1970079 doi 10 2307 1970079 Mehta M L Random Matrices Amsterdam Elsevier Academic Press 2004 ISBN 0 12 088409 7 Bohigas O Giannoni M J Schmit Schmit Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws Phys Rev Lett 1984 52 1 1 4 Bibcode 1984PhRvL 52 1B doi 10 1103 PhysRevLett 52 1 Aaronson Scott Arkhipov Alex The computational complexity of linear optics Theory of Computing 2013 9 143 252 doi 10 4086 toc 2013 v009a004 Russell Nicholas Chakhmakhchyan Levon O Brien Jeremy Laing Anthony Direct dialling of Haar random unitary matrices New J Phys 2017 19 3 033007 Bibcode 2017NJPh 19c3007R arXiv 1506 06220 doi 10 1088 1367 2630 aa60ed Random Matrix Theory and Chiral Symmetry in QCD Annu Rev Nucl Part Sci 2000 50 343 410 Bibcode 2000ARNPS 50 343V arXiv hep ph 0003017 doi 10 1146 annurev nucl 50 1 343 Horizon in random matrix theory the Hawking radiation and flow of cold atoms Phys Rev Lett October 2009 103 16 166401 Bibcode 2009PhRvL 103p6401F PMID 19905710 arXiv 0905 3533 doi 10 1103 PhysRevLett 103 166401 Magnetic field asymmetry of nonlinear mesoscopic transport Phys Rev Lett September 2004 93 10 106802 Bibcode 2004PhRvL 93j6802S PMID 15447435 arXiv cond mat 0404387 doi 10 1103 PhysRevLett 93 106802 Spin torque and waviness in magnetic multilayers a bridge between Valet Fert theory and quantum approaches Phys Rev Lett August 2009 103 6 066602 Bibcode 2009PhRvL 103f6602R PMID 19792592 arXiv 0902 4360 doi 10 1103 PhysRevLett 103 066602 Callaway DJE Random matrices fractional statistics and the quantum Hall effect Phys Rev B April 1991 43 10 8641 8643 Bibcode 1991PhRvB 43 8641C PMID 9996505 doi 10 1103 PhysRevB 43 8641 Correlated random band matrices localization delocalization transitions Phys Rev E June 2000 61 6 Pt A 6278 86 Bibcode 2000PhRvE 61 6278J PMID 11088301 arXiv cond mat 9911467 doi 10 1103 PhysRevE 61 6278 Spin orbit coupling antilocalization and parallel magnetic fields in quantum dots Phys Rev Lett December 2002 89 27 276803 Bibcode 2002PhRvL 89A6803Z PMID 12513231 arXiv cond mat 0208436 doi 10 1103 PhysRevLett 89 276803 Bahcall SR Random Matrix Model for Superconductors in a Magnetic Field Phys Rev Lett December 1996 77 26 5276 5279 Bibcode 1996PhRvL 77 5276B PMID 10062760 arXiv cond mat 9611136 doi 10 1103 PhysRevLett 77 5276 Keating Jon The Riemann zeta function and quantum chaology Proc Internat School of Phys Enrico Fermi 1993 CXIX 145 185 ISBN 9780444815880 doi 10 1016 b978 0 444 81588 0 50008 0 Wishart J Generalized product moment distribution in samples Biometrika 1928 20A 1 2 32 52 doi 10 1093 biomet 20a 1 2 32 Tropp J User Friendly Tail Bounds for Sums of Random Matrices Foundations of Computational Mathematics 2011 12 4 389 434 arXiv 1004 4389 doi 10 1007 s10208 011 9099 z von Neumann J Goldstine H H Numerical inverting of matrices of high order Bull Amer Math Soc 1947 53 11 1021 1099 doi 10 1090 S0002 9904 1947 08909 6 Edelman A Rao N R Random matrix theory Acta Numerica 2005 14 233 297 Bibcode 2005AcNum 14 233E doi 10 1017 S0962492904000236 Chow Gregory P Analysis and Control of Dynamic Economic Systems New York Wiley 1976 ISBN 0 471 15616 7 Turnovsky Stephen Optimal stabilization policies for stochastic linear systems The case of correlated multiplicative and additive disturbances Review of Economic Studies 1976 43 1 191 194 JSTOR 2296741 doi 10 2307 2296614 22 0 22 1 Turnovsky Stephen The stability properties of optimal economic policies American Economic Review 1974 64 1 136 148 JSTOR 1814888 Garcia del Molino Luis Carlos Pakdaman Khashayar Touboul Jonathan Wainrib Gilles Synchronization in random balanced networks Physical Review E October 2013 88 4 042824 Bibcode 2013PhRvE 88d2824G arXiv 1306 2576 doi 10 1103 PhysRevE 88 042824 Rajan Kanaka Abbott L Eigenvalue Spectra of Random Matrices for Neural Networks Physical Review Letters November 2006 97 18 188104 Bibcode 2006PhRvL 97r8104R PMID 17155583 doi 10 1103 PhysRevLett 97 188104 Wainrib Gilles Touboul Jonathan Topological and Dynamical Complexity of Random Neural Networks Physical Review Letters March 2013 110 11 118101 Bibcode 2013PhRvL 110k8101W PMID 25166580 arXiv 1210 5082 doi 10 1103 PhysRevLett 110 118101 Muir Dylan Mrsic Flogel Thomas Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks PDF Phys Rev E 2015 91 4 042808 2020 01 13 Bibcode 2015PhRvE 91d2808M PMID 25974548 doi 10 1103 PhysRevE 91 042808 原始内容 PDF 存档于2018 07 21 Cosme Louart Zhenyu Liao and Romain Couillet A RANDOM MATRIX APPROACH TO NEURAL NETWORKS PDF 原始内容 PDF 存档于2020 01 13 Zhenyu Liao Romain Couillet The Dynamics of Learning A Random Matrix Approach PDF 原始内容 PDF 存档于2020 11 12 Jeffrey Pennington Pratik Worah Nonlinear random matrix theory for deep learning PDF 原始内容 PDF 存档于2020 11 03 Mehta M L Random Matrices Amsterdam Elsevier Academic Press 2004 ISBN 0 12 088409 7 Anderson G W Guionnet A Zeitouni O An introduction to random matrices Cambridge Cambridge University Press 2010 ISBN 978 0 521 19452 5 Akemann G Baik J Di Francesco P The Oxford Handbook of Random Matrix Theory Oxford Oxford University Press 2011 ISBN 978 0 19 957400 1 Edelman A Rao N R Random matrix theory Acta Numerica 2005 14 233 297 Bibcode 2005AcNum 14 233E doi 10 1017 S0962492904000236 Pastur L A Spectra of random self adjoint operators Russ Math Surv 1973 28 1 1 67 Bibcode 1973RuMaS 28 1P doi 10 1070 RM1973v028n01ABEH001396 Diaconis Persi Patterns in eigenvalues the 70th Josiah Willard Gibbs lecture American Mathematical Society Bulletin New Series 2003 40 2 155 178 MR 1962294 doi 10 1090 S0273 0979 03 00975 3 Diaconis Persi What is a random matrix Notices of the American Mathematical Society 2005 52 11 1348 1349 2020 01 13 ISSN 0002 9920 MR 2183871 原始内容存档于2019 03 28 Wigner E Characteristic vectors of bordered matrices with infinite dimensions Annals of Mathematics 1955 62 3 548 564 JSTOR 1970079 doi 10 2307 1970079 Wishart J Generalized product moment distribution in samples Biometrika 1928 20A 1 2 32 52 doi 10 1093 biomet 20a 1 2 32 von Neumann J Goldstine H H Numerical inverting of matrices of high order Bull Amer Math Soc 1947 53 11 1021 1099 doi 10 1090 S0002 9904 1947 08909 6 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