随机矩阵

在概率论和数学物理中，随机矩阵（英语：Random matrix）是一个矩阵值的随机变量，也就是说，一个矩阵中的所有元素都是随机变量。^[1]

应用

物理

原子核物理学^[2]^[3]，量子场论
量子混沌（quantum chaos）Bohigas–Giannoni–Schmit（BGS）猜想^[4]
量子光学^[5]^[6]
杨-米尔斯理论（量子色动力学）^[7]
两维的量子引力，AdS/CFT对偶，^[8]
介观物理学,^[9]
自旋转移矩,^[10]
小数量子霍尔效果,^[11]
安德森的本地化（Anderson localization）^[12]
量子点,^[13]
超导现象^[14]

其他（AI、数学、统计）

随机矩阵模型

设 $H$ 是 $N\times N$ 的矩阵，有下面的概率测度：

$P(H)={\frac {1}{Z}}e^{-Ntr(V(H))}$

例子，高斯模型： $V(H)=H^{2}/2$ 。

GUE (Gaussian Unitary Ensemble)：H是埃尔米特矩阵。通过1/N展开，维格纳半圆分布描述H的大N特征值的几率密度函数 $\rho (E)$ 。^[1]
GOE (Orthogonal)：H是对称矩阵
GSE (Symplectic)：H是四元数的矩阵（Quaternion matrix）

参见

维格纳半圆分布
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

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）)
徐一鸿的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)

参考文献

^ ^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.
^ 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.
^ 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.

[:0-1] 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-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.

[mehta-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.

[wishart-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.

[vng-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.

[er-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.

[#1-22] 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. 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）.

[27] 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）.

[28] Zhenyu Liao, Romain Couillet. The Dynamics of Learning: A Random Matrix Approach (PDF). （原始内容 (PDF)存档于2020-11-12）.

[29] Jeffrey Pennington, Pratik Worah. Nonlinear random matrix theory for deep learning (PDF). （原始内容 (PDF)存档于2020-11-03）.

[mehta2-30] 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.

[er2-33] Edelman, A.; Rao, N.R. Random matrix theory. Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236.

[pastur72-34] 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. 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）.

[wigner2-37] Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079.

[wishart2-38] Wishart, J. Generalized product moment distribution in samples. Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32.

[vng2-39] 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.

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