随机矩阵
外观
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
[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ζ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]
随机矩阵模型
[编辑]设是的矩阵,有下面的概率测度:
例子,高斯模型:。
- 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。[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·戴森气体模型(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://enbaike.710302.xyz/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
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- ^ 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
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- ^ 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
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- ^ 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
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- ^ 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.
- ^ 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
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- ^ 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
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- ^ 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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