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.^{[1]} 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.^{[2]} 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.^{[3]}

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).^{[4]} 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).^{[5]}

Random matrix theory has also found applications to the chiral Dirac operator in quantum chromodynamics,^{[6]} quantum gravity in two dimensions,^{[7]} mesoscopic physics,^{[8]} spin-transfer torque,^{[9]} the fractional quantum Hall effect,^{[10]} Anderson localization,^{[11]} quantum dots,^{[12]} and superconductors^{[13]}

In multivariate statistics, random matrices were introduced by John Wishart, who sought to estimate covariance matrices of large samples.^{[14]} Chernoff-, Bernstein-, and Hoeffding-type inequalities can typically be strengthened when applied to the maximal eigenvalue of a finite sum of random Hermitian matrices.^{[15]}

In numerical analysis, random matrices have been used since the work of John von Neumann and Herman Goldstine^{[16]} 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.^{[17]}

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.^{[18]} 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^{[19]} 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^{[20]}^{[21]} and time to synchrony depends on network topology.^{[22]}^{[23]}

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.^{[24]}^{: ch. 13 }^{[25]}^{[26]} 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.

The most-commonly studied random matrix distributions are the Gaussian ensembles.

The **Gaussian unitary ensemble** is described by the Gaussian measure with density

on the space of Hermitian matrices . Here
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* = (*H*_{ij})^{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* = (*H*_{ij})^{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.

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 ⟨*H*_{ij}⟩ = 0, and two-point correlations given by

- ,

from which all higher correlations follow by Isserlis' theorem.

The joint probability density for the eigenvalues *λ*_{1},*λ*_{2},...,*λ*_{n} of GUE/GOE/GSE is given by

where *Z*_{β,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.^{[27]}

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,

for .

*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.

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 .

The *empirical spectral measure* *μ _{H}* of

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.^{[28]}^{[29]}

The limit of the empirical spectral measure of invariant matrix ensembles is described by a certain integral equation which arises from potential theory.^{[30]}

For the linear statistics *N*_{f,H} = *n*^{−1} Σ *f*(*λ*_{j}), 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

is known.^{[31]}^{[32]}

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.

Formally, fix in the interior of the support of . Then consider the point process

where are the eigenvalues of the random matrix.

The point process captures the statistical properties of eigenvalues in the vicinity of . For the Gaussian ensembles, the limit of is known;^{[2]} thus, for GUE it is a determinantal point process with the kernel

(the *sine 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^{[33]}^{[34]} and Wigner matrices.^{[35]}^{[36]}

Main article: Tracy–Widom distribution |

The joint probability density of the eigenvalues of random Hermitian matrices , with partition functions of the form

where

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.

**Theorem** [Dyson-Mehta]
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

Main article: Wishart distribution |

*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^{[28]} by Vladimir Marchenko and Leonid Pastur.

*See circular ensembles.*

*See circular law.*