In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.
In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation.
In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts that the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behaviour are described by random matrix theory.
In quantum optics, transformations described by random unitary matrices are crucial for demonstrating the advantage of quantum over classical computation (see, e.g., the boson sampling model). Moreover, such random unitary transformations can be directly implemented in an optical circuit, by mapping their parameters to optical circuit components (that is beam splitters and phase shifters).
Random matrix theory has also found applications to the chiral Dirac operator in quantum chromodynamics, quantum gravity in two dimensions, mesoscopic physics, spin-transfer torque, the fractional quantum Hall effect, Anderson localization, quantum dots, and superconductors
Mathematical statistics and numerical analysis
In multivariate statistics, random matrices were introduced by John Wishart, who sought to estimate covariance matrices of large samples. Chernoff-, Bernstein-, and Hoeffding-type inequalities can typically be strengthened when applied to the maximal eigenvalue of a finite sum of random Hermitian matrices.
In numerical analysis, random matrices have been used since the work of John von Neumann and Herman Goldstine to describe computation errors in operations such as matrix multiplication. Although random entries are traditional "generic" inputs to an algorithm, the concentration of measure associated with random matrix distributions implies that random matrices will not test large portions of an algorithm's input space.
In number theory, the distribution of zeros of the Riemann zeta function (and other L-functions) is modeled by the distribution of eigenvalues of certain random matrices. The connection was first discovered by Hugh Montgomery and Freeman J. Dyson. It is connected to the Hilbert–Pólya conjecture.
In the field of theoretical neuroscience, random matrices are increasingly used to model the network of synaptic connections between neurons in the brain. Dynamical models of neuronal networks with random connectivity matrix were shown to exhibit a phase transition to chaos when the variance of the synaptic weights crosses a critical value, at the limit of infinite system size. Results on random matrices have also shown that the dynamics of random-matrix models are insensitive to mean connection strength. Instead, the stability of fluctuations depends on connection strength variation and time to synchrony depends on network topology.
In optimal control theory, the evolution of n state variables through time depends at any time on their own values and on the values of k control variables. With linear evolution, matrices of coefficients appear in the state equation (equation of evolution). In some problems the values of the parameters in these matrices are not known with certainty, in which case there are random matrices in the state equation and the problem is known as one of stochastic control.: ch. 13  A key result in the case of linear-quadratic control with stochastic matrices is that the certainty equivalence principle does not apply: while in the absence of multiplier uncertainty (that is, with only additive uncertainty) the optimal policy with a quadratic loss function coincides with what would be decided if the uncertainty were ignored, the optimal policy may differ if the state equation contains random coefficients.
In computational mechanics, epistemic uncertainties underlying the lack of knowledge about the physics of the modeled system give rise to mathematical operators associated with the computational model, which are deficient in a certain sense. Such operators lack certain properties linked to unmodeled physics. When such operators are discretized to perform computational simulations, their accuracy is limited by the missing physics. To compensate for this deficiency of the mathematical operator, it is not enough to make the model parameters random, it is necessary to consider a mathematical operator that is random and can thus generate families of computational models in the hope that one of these captures the missing physics. Random matrices have been used in this sense, with applications in vibroacoustics, wave propagations, materials science, fluid mechanics, heat transfer, etc.
The most-commonly studied random matrix distributions are the Gaussian ensembles.
The Gaussian unitary ensemble is described by the Gaussian measure with density
The Gaussian orthogonal ensemble is described by the Gaussian measure with density
i,j=1. Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry.
The Gaussian symplectic ensemble is described by the Gaussian measure with density
i,j=1. Its distribution is invariant under conjugation by the symplectic group, and it models Hamiltonians with time-reversal symmetry but no rotational symmetry.
The Gaussian ensembles GOE, GUE and GSE are often denoted by their Dyson index, β = 1 for GOE, β = 2 for GUE, and β = 4 for GSE. This index counts the number of real components per matrix element. The ensembles as defined here have Gaussian distributed matrix elements with mean ⟨Hij⟩ = 0, and two-point correlations given by
where Zβ,n is a normalization constant which can be explicitly computed, see Selberg integral. In the case of GUE (β = 2), the formula (1) describes a determinantal point process. Eigenvalues repel as the joint probability density has a zero (of th order) for coinciding eigenvalues .
For the distribution of the largest eigenvalue for GOE, GUE and Wishart matrices of finite dimensions, see.
Distribution of level spacings
From the ordered sequence of eigenvalues , one defines the normalized spacings , where is the mean spacing. The probability distribution of spacings is approximately given by,
The numerical constants are such that is normalized:
Wigner matrices are random Hermitian matrices such that the entries
Invariant matrix ensembles are random Hermitian matrices with density on the space of real symmetric/Hermitian/quaternionic Hermitian matrices, which is of the form where the function V is called the potential.
The Gaussian ensembles are the only common special cases of these two classes of random matrices.
Spectral theory of random matrices
The spectral theory of random matrices studies the distribution of the eigenvalues as the size of the matrix goes to infinity.
In the global regime, one is interested in the distribution of linear statistics of the form .
Empirical spectral measure
The empirical spectral measure μH of H is defined by
Usually, the limit of is a deterministic measure; this is a particular case of self-averaging. The cumulative distribution function of the limiting measure is called the integrated density of states and is denoted N(λ). If the integrated density of states is differentiable, its derivative is called the density of states and is denoted ρ(λ).
The limit of the empirical spectral measure for Wigner matrices was described by Eugene Wigner; see Wigner semicircle distribution and Wigner surmise. As far as sample covariance matrices are concerned, a theory was developed by Marčenko and Pastur.
For the linear statistics Nf,H = n−1 Σ f(λj), one is also interested in the fluctuations about ∫ f(λ) dN(λ). For many classes of random matrices, a central limit theorem of the form
In the local regime, one is interested in the spacings between eigenvalues, and, more generally, in the joint distribution of eigenvalues in an interval of length of order 1/n. One distinguishes between bulk statistics, pertaining to intervals inside the support of the limiting spectral measure, and edge statistics, pertaining to intervals near the boundary of the support.
The point process captures the statistical properties of eigenvalues in the vicinity of . For the Gaussian ensembles, the limit of is known; thus, for GUE it is a determinantal point process with the kernel
The universality principle postulates that the limit of as should depend only on the symmetry class of the random matrix (and neither on the specific model of random matrices nor on ). Rigorous proofs of universality are known for invariant matrix ensembles and Wigner matrices.
The joint probability density of the eigenvalues of random Hermitian matrices , with partition functions of the form
The following result expresses these correlation functions as determinants of the matrices formed from evaluating the appropriate integral kernel at the pairs of points appearing within the correlator.
Theorem [Dyson-Mehta] For any , the -point correlation function can be written as a determinant
Other classes of random matrices
Wishart matrices are n × n random matrices of the form H = X X*, where X is an n × m random matrix (m ≥ n) with independent entries, and X* is its conjugate transpose. In the important special case considered by Wishart, the entries of X are identically distributed Gaussian random variables (either real or complex).
The limit of the empirical spectral measure of Wishart matrices was found by Vladimir Marchenko and Leonid Pastur.
Random unitary matrices
Non-Hermitian random matrices
- Mehta, M.L. (2004). Random Matrices. Amsterdam: Elsevier/Academic Press. ISBN 0-12-088409-7.
- Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. ISBN 978-0-521-19452-5.
- Akemann, G.; Baik, J.; Di Francesco, P. (2011). The Oxford Handbook of Random Matrix Theory. Oxford: Oxford University Press. ISBN 978-0-19-957400-1.
- Edelman, A.; Rao, N.R (2005). "Random matrix theory". Acta Numerica. 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236. S2CID 16038147.
- Pastur, L.A. (1973). "Spectra of random self-adjoint operators". Russ. Math. Surv. 28 (1): 1–67. Bibcode:1973RuMaS..28....1P. doi:10.1070/RM1973v028n01ABEH001396. S2CID 250796916.
- Diaconis, Persi (2003). "Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture". Bulletin of the American Mathematical Society. New Series. 40 (2): 155–178. doi:10.1090/S0273-0979-03-00975-3. MR 1962294.
- Diaconis, Persi (2005). "What is ... a random matrix?". Notices of the American Mathematical Society. 52 (11): 1348–1349. ISSN 0002-9920. MR 2183871.
- Eynard, Bertrand; Kimura, Taro; Ribault, Sylvain (2015-10-15). "Random matrices". arXiv:1510.04430v2 [math-ph].
- Wigner, E. (1955). "Characteristic vectors of bordered matrices with infinite dimensions". Annals of Mathematics. 62 (3): 548–564. doi:10.2307/1970079. JSTOR 1970079.
- Wishart, J. (1928). "Generalized product moment distribution in samples". Biometrika. 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32.
- von Neumann, J.; Goldstine, H.H. (1947). "Numerical inverting of matrices of high order". Bull. Amer. Math. Soc. 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6.
- Wigner 1955
- Mehta 2004
- Bohigas, O.; Giannoni, M.J.; Schmit, Schmit (1984). "Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws". Phys. Rev. Lett. 52 (1): 1–4. Bibcode:1984PhRvL..52....1B. doi:10.1103/PhysRevLett.52.1.
- Aaronson, Scott; Arkhipov, Alex (2013). "The computational complexity of linear optics". Theory of Computing. 9: 143–252. doi:10.4086/toc.2013.v009a004.
- Russell, Nicholas; Chakhmakhchyan, Levon; O'Brien, Jeremy; Laing, Anthony (2017). "Direct dialling of Haar random unitary matrices". New J. Phys. 19 (3): 033007. arXiv:1506.06220. Bibcode:2017NJPh...19c3007R. doi:10.1088/1367-2630/aa60ed. S2CID 46915633.
- Verbaarschot JJ, Wettig T (2000). "Random Matrix Theory and Chiral Symmetry in QCD". Annu. Rev. Nucl. Part. Sci. 50: 343–410. arXiv:hep-ph/0003017. Bibcode:2000ARNPS..50..343V. doi:10.1146/annurev.nucl.50.1.343. S2CID 119470008.
- Franchini F, Kravtsov VE (October 2009). "Horizon in random matrix theory, the Hawking radiation, and flow of cold atoms". Phys. Rev. Lett. 103 (16): 166401. arXiv:0905.3533. Bibcode:2009PhRvL.103p6401F. doi:10.1103/PhysRevLett.103.166401. PMID 19905710. S2CID 11122957.
- Sánchez D, Büttiker M (September 2004). "Magnetic-field asymmetry of nonlinear mesoscopic transport". Phys. Rev. Lett. 93 (10): 106802. arXiv:cond-mat/0404387. Bibcode:2004PhRvL..93j6802S. doi:10.1103/PhysRevLett.93.106802. PMID 15447435. S2CID 11686506.
- Rychkov VS, Borlenghi S, Jaffres H, Fert A, Waintal X (August 2009). "Spin torque and waviness in magnetic multilayers: a bridge between Valet-Fert theory and quantum approaches". Phys. Rev. Lett. 103 (6): 066602. arXiv:0902.4360. Bibcode:2009PhRvL.103f6602R. doi:10.1103/PhysRevLett.103.066602. PMID 19792592. S2CID 209013.
- Callaway DJE (April 1991). "Random matrices, fractional statistics, and the quantum Hall effect". Phys. Rev. B. 43 (10): 8641–8643. Bibcode:1991PhRvB..43.8641C. doi:10.1103/PhysRevB.43.8641. PMID 9996505.
- Janssen M, Pracz K (June 2000). "Correlated random band matrices: localization-delocalization transitions". Phys. Rev. E. 61 (6 Pt A): 6278–86. arXiv:cond-mat/9911467. Bibcode:2000PhRvE..61.6278J. doi:10.1103/PhysRevE.61.6278. PMID 11088301. S2CID 34140447.
- Zumbühl DM, Miller JB, Marcus CM, Campman K, Gossard AC (December 2002). "Spin-orbit coupling, antilocalization, and parallel magnetic fields in quantum dots". Phys. Rev. Lett. 89 (27): 276803. arXiv:cond-mat/0208436. Bibcode:2002PhRvL..89A6803Z. doi:10.1103/PhysRevLett.89.276803. PMID 12513231. S2CID 9344722.
- Bahcall SR (December 1996). "Random Matrix Model for Superconductors in a Magnetic Field". Phys. Rev. Lett. 77 (26): 5276–5279. arXiv:cond-mat/9611136. Bibcode:1996PhRvL..77.5276B. doi:10.1103/PhysRevLett.77.5276. PMID 10062760. S2CID 206326136.
- Wishart 1928
- Tropp, J. (2011). "User-Friendly Tail Bounds for Sums of Random Matrices". Foundations of Computational Mathematics. 12 (4): 389–434. arXiv:1004.4389. doi:10.1007/s10208-011-9099-z. S2CID 17735965.
- von Neumann & Goldstine 1947
- Edelman & Rao 2005
- Keating, Jon (1993). "The Riemann zeta-function and quantum chaology". Proc. Internat. School of Phys. Enrico Fermi. CXIX: 145–185. doi:10.1016/b978-0-444-81588-0.50008-0. ISBN 9780444815880.
- Sompolinsky, H.; Crisanti, A.; Sommers, H. (July 1988). "Chaos in Random Neural Networks". Physical Review Letters. 61 (3): 259–262. Bibcode:1988PhRvL..61..259S. doi:10.1103/PhysRevLett.61.259. PMID 10039285.
- Rajan, Kanaka; Abbott, L. (November 2006). "Eigenvalue Spectra of Random Matrices for Neural Networks". Physical Review Letters. 97 (18): 188104. Bibcode:2006PhRvL..97r8104R. doi:10.1103/PhysRevLett.97.188104. PMID 17155583.
- Wainrib, Gilles; Touboul, Jonathan (March 2013). "Topological and Dynamical Complexity of Random Neural Networks". Physical Review Letters. 110 (11): 118101. arXiv:1210.5082. Bibcode:2013PhRvL.110k8101W. doi:10.1103/PhysRevLett.110.118101. PMID 25166580. S2CID 1188555.
- Timme, Marc; Wolf, Fred; Geisel, Theo (February 2004). "Topological Speed Limits to Network Synchronization". Physical Review Letters. 92 (7): 074101. arXiv:cond-mat/0306512. Bibcode:2004PhRvL..92g4101T. doi:10.1103/PhysRevLett.92.074101. PMID 14995853. S2CID 5765956.
- Muir, Dylan; Mrsic-Flogel, Thomas (2015). "Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks" (PDF). Phys. Rev. E. 91 (4): 042808. Bibcode:2015PhRvE..91d2808M. doi:10.1103/PhysRevE.91.042808. PMID 25974548.
- Chow, Gregory P. (1976). Analysis and Control of Dynamic Economic Systems. New York: Wiley. ISBN 0-471-15616-7.
- Turnovsky, Stephen (1976). "Optimal stabilization policies for stochastic linear systems: The case of correlated multiplicative and additive disturbances". Review of Economic Studies. 43 (1): 191–194. doi:10.2307/2296614. JSTOR 2296741.
- Turnovsky, Stephen (1974). "The stability properties of optimal economic policies". American Economic Review. 64 (1): 136–148. JSTOR 1814888.
- Soize, C. (2000-07-01). "A nonparametric model of random uncertainties for reduced matrix models in structural dynamics". Probabilistic Engineering Mechanics. 15 (3): 277–294. doi:10.1016/S0266-8920(99)00028-4. ISSN 0266-8920.
- Soize, C. (2005-04-08). "Random matrix theory for modeling uncertainties in computational mechanics". Computer methods in applied mechanics and engineering. 194 (12–16): 1333–1366. doi:10.1016/j.cma.2004.06.038. ISSN 1879-2138.
- Chiani M (2014). "Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution". Journal of Multivariate Analysis. 129: 69–81. arXiv:1209.3394. doi:10.1016/j.jmva.2014.04.002. S2CID 15889291.
- .Marčenko, V A; Pastur, L A (1967). "Distribution of eigenvalues for some sets of random matrices". Mathematics of the USSR-Sbornik. 1 (4): 457–483. Bibcode:1967SbMat...1..457M. doi:10.1070/SM1967v001n04ABEH001994.
- Pastur 1973
- Pastur, L.; Shcherbina, M. (1995). "On the Statistical Mechanics Approach in the Random Matrix Theory: Integrated Density of States". J. Stat. Phys. 79 (3–4): 585–611. Bibcode:1995JSP....79..585D. doi:10.1007/BF02184872. S2CID 120731790.
- Johansson, K. (1998). "On fluctuations of eigenvalues of random Hermitian matrices". Duke Math. J. 91 (1): 151–204. doi:10.1215/S0012-7094-98-09108-6.
- Pastur, L.A. (2005). "A simple approach to the global regime of Gaussian ensembles of random matrices". Ukrainian Math. J. 57 (6): 936–966. doi:10.1007/s11253-005-0241-4. S2CID 121531907.
- Pastur, L.; Shcherbina, M. (1997). "Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles". Journal of Statistical Physics. 86 (1–2): 109–147. Bibcode:1997JSP....86..109P. doi:10.1007/BF02180200. S2CID 15117770.
- Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X. (1997). "Asymptotics for polynomials orthogonal with respect to varying exponential weights". International Mathematics Research Notices. 1997 (16): 759–782. doi:10.1155/S1073792897000500.
- Erdős, L.; Péché, S.; Ramírez, J.A.; Schlein, B.; Yau, H.T. (2010). "Bulk universality for Wigner matrices". Communications on Pure and Applied Mathematics. 63 (7): 895–925.
- Tao, Terence; Vu, Van H. (2010). "Random matrices: universality of local eigenvalue statistics up to the edge". Communications in Mathematical Physics. 298 (2): 549–572. arXiv:0908.1982. Bibcode:2010CMaPh.298..549T. doi:10.1007/s00220-010-1044-5. S2CID 16594369.