Type or paste a DOI name into the text box. In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. In the field of theoretical neuroscience, random matrices are increasingly used to model the network of synaptic connections between neurons in the brain. In optimal control theory, the evolution of n state variables through time depends at any time on their the oxford handbook of random matrix theory pdf values and on the values of k control variables.

The most studied random matrix ensembles are the Gaussian ensembles. 2 is a normalization constant, chosen so that the integral of the density is equal to one. The term unitary refers to the fact that the distribution is invariant under unitary conjugation. The Gaussian unitary ensemble models Hamiltonians lacking time-reversal symmetry. Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry. Its distribution is invariant under conjugation by the symplectic group, and it models Hamiltonians with time-reversal symmetry but no rotational symmetry. Zβ,n is a normalization constant which can be explicitly computed, see Selberg integral.

For the distribution of the largest eigenvalue for GOE, GUE and Wishart matrices of finite dimensions, see. The Gaussian ensembles are the only common special cases of these two classes of random matrices. The spectral theory of random matrices studies the distribution of the eigenvalues as the size of the matrix goes to infinity. As far as sample covariance matrices are concerned, a theory was developed by Marčenko and Pastur.

The limit of the empirical spectral measure of invariant matrix ensembles is described by a certain integral equation which arises from potential theory. 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. This was rigorously proved for several models of random matrices: for invariant matrix ensembles, for Wigner matrices, et cet. Characteristic vectors of bordered matrices with infinite dimensions”. Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws”. The computational complexity of linear optics”.

Direct dialling of Haar random unitary matrices”. Random Matrix Theory and Chiral Symmetry in QCD”. Horizon in random matrix theory, the Hawking radiation, and flow of cold atoms”. Magnetic-field asymmetry of nonlinear mesoscopic transport”. Spin torque and waviness in magnetic multilayers: a bridge between Valet-Fert theory and quantum approaches”. Random matrices, fractional statistics, and the quantum Hall effect”. Correlated random band matrices: localization-delocalization transitions”.

Spin-orbit coupling, antilocalization, and parallel magnetic fields in quantum dots”. Random Matrix Model for Superconductors in a Magnetic Field”. Generalized product moment distribution in samples”. User-Friendly Tail Bounds for Sums of Random Matrices”. Numerical inverting of matrices of high order”. The Riemann zeta-function and quantum chaology”. Eigenvalue Spectra of Random Matrices for Neural Networks”.

Topological and Dynamical Complexity of Random Neural Networks”. Topological Speed Limits to Network Synchronization”. Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks”. Analysis and Control of Dynamic Economic Systems. Optimal stabilization policies for stochastic linear systems: The case of correlated multiplicative and additive disturbances”. The stability properties of optimal economic policies”.

Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution”. Distribution of eigenvalues for some sets of random matrices”. On the Statistical Mechanics Approach in the Random Matrix Theory: Integrated Density of States”. On fluctuations of eigenvalues of random Hermitian matrices”. A simple approach to the global regime of Gaussian ensembles of random matrices”. Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles”.

Asymptotics for polynomials orthogonal with respect to varying exponential weights”. Communications on Pure and Applied Mathematics. Random matrices: universality of local eigenvalue statistics up to the edge”. The Oxford Handbook of Random Matrix Theory.