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).
The relation of free probability with random matrices is a key reason for the wide use of free probability in other subjects. Voiculescu introduced the concept of freeness around 1983 in an operator algebraic context; at the beginning there was no relation at all with random matrices. This connection was only revealed later in 1991 by Voiculescu; he was motivated by the fact that the limit distribution which he found in his free central limit theorem had appeared before in Wigner's semi-circle law in the random matrix context.
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: GOE, GUE and GSE. They 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.
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.
The Gaussian orthogonal ensemble is described by the Gaussian measure with density
on the space of n × n real symmetric matrices H = (Hij)n 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
on the space of n × n Hermitian quaternionic matrices, e.g. symmetric square matrices composed of quaternions, H = (Hij)n 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.
Point correlation functions
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 .
The distribution of the largest eigenvalue for GOE, and GUE, are explicitly solvable. They converge to the Tracy–Widom distribution after shifting and scaling appropriately.
Convergence to Wigner semicircular distribution
The spectrum, divided by , converges in distribution to the semicircular distribution . Here is the variance of off-diagonal entries.
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,
for the orthogonal ensemble GOE ,
for the unitary ensemble GUE , and
for the symplectic ensemble GSE .
The numerical constants are such that is normalized:
and the mean spacing is,
Wigner matrices are random Hermitian matrices such that the entries
above the main diagonal are independent random variables with zero mean and have identical second moments.
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. This is a consequence of a theorem by Porter and Rosenzweig.
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 measureμH of H is defined by
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 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
and is the standard Lebesgue measure on the space of Hermitian matrices, is given by
The -point correlation functions (or marginal distributions)
are defined as
which are skew symmetric functions of their variables.
In particular, the one-point correlation function, or density of states, is
Its integral over a Borel set gives the expected number of eigenvalues contained in :
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.
For any , the -point correlation function can be written as a determinant
where is the th Christoffel-Darboux kernel
associated to , written in terms of the quasipolynomials
where is a complete sequence of monic polynomials, of the degrees indicated, satisfying the orthogonilty conditions
Wishart matrices are n × n random matrices of the form H = XX*, 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).