random matrix in nLab

Contents

Contents

Idea

A random matrix is a matrix-valued random variable. Random matrix theory studies mainly the behaviour of eigenvalues and various functions of random matrices; as such it has large importance in physics.

• Calogero-Moser system,

• large N limit

References

General

Review:

• Leonid Petrov, Random Matrices, lecture notes 2019 (pdf slides, pdf, webpage)

• Madan Lal Mehta, Random matrices, 3rd ed. Pure and Applied Math. (Amsterdam) 142, Elsevier/Academic Press 2004. xviii+688 pp. MR2129906, gBooks

• Wikipedia, Random matrix

• Terrence Tao?, Topics in random matrix theory pdf

See also:

• V. L. Girko, Theory of random determinants, Transl. from Russian (Višča Škola, Kiev 1980, MR82h:60002) Mathematics and its Applications (Soviet Series) 45, Kluwer 1990, MR1080966

• Freeman Dyson, Statistical theory of the energy levels of complex systems, I, J. Math. Phys. 3 1962 140–156, MR143556, doi; II, JMP 3 1962 157–165, MR143557, doi; III, JMP 3 1962 166–175, MR143558, doi; A Brownian-motion model for the eigenvalues of a random matrix, JMP 3 1962 1191–1198, MR148397, doi; Fredholm determinants and inverse scattering problems, Comm. Math. Phys. 47, 171–183 (1976) MR406201 euclid

• J J M Verbaarschot, M R Zirnbauer, Critique of the replica trick, J. Phys. A: Math. Gen. 17 (1985) 1093-1109, pdf

• Patrik L. Ferrari, Why random matrices share universal processes with interacting particle systems?, arxiv/1312.1126

• Bertrand Eynard, Taro Kimura, Sylvain Ribault, Random matrices, lecture notes (arxiv/1510.04430)

• Greg W. Anderson, Alice Guionnet, Ofer Zeitouni, An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics

In string/M-theory

Random matrix theory applies to black holes in string theory:

via the SYK model:

• Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, Joseph Polchinski, Phil Saad, Stephen Shenker, Douglas Stanford, Alexandre Streicher, Masaki Tezuka, Black Holes and Random Matrices, JHEP 1705:118, 2017 (arXiv:1611.04650)

• Yiyang Jia, Jacobus J. M. Verbaarschot, Spectral Fluctuations in the Sachdev-Ye-Kitaev Model (arXiv:1912.11923)

via the BFSS matrix model:

• Haoxing Du, Vatche Sahakian, Emergent geometry from stochastic dynamics, or Hawking evaporation in M(atrix) theory (arXiv:1812.05020)

via AdS/CFT for Jackiw-Teitelboim gravity:

• Phil Saad, Stephen Shenker, Douglas Stanford, JT gravity as a matrix integral (arXiv:1903.11115)

• Douglas Stanford, Edward Witten, JT Gravity and the Ensembles of Random Matrix Theory (arXiv:1907.03363)