Random matrix

# Random matrix

Discussion

Encyclopedia
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, a random matrix is a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

-valued random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

. Many important properties of physical system
Physical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...

s can be represented mathematically as matrix problems. For example, the thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

of a lattice
Lattice model (physics)
In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...

can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

### Physics

In nuclear physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...

, random matrices were introduced by Eugene Wigner to model the spectra of heavy atoms. He postulated that the spacings between the lines in the spectrum of a heavy atom 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
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...

, random matrices model the behaviour of large disordered Hamiltonians
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

in the mean field approximation.

In quantum chaos
Quantum chaos
Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is, "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle...

, 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.

Random matrix theory has also found applications to quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

in two dimensions, mesoscopic physics, and more
.

### Mathematical statistics and numerical analysis

In multivariate statistics
Multivariate statistics
Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis...

, random matrices were introduced by John Wishart
John Wishart (statistician)
John Wishart was a Scottish mathematician and agricultural statistician.He worked successively at University College London with Karl Pearson, at Rothamsted Experimental Station with Ronald Fisher, and then as a reader in statistics in the University of Cambridge where he became the first...

for statistical analysis of large samples; see estimation of covariance matrices
Estimation of covariance matrices
In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution...

.

In numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, random matrices have been used since the work of John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

and Herman Goldstine to describe computation errors in operations such as matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

### Number theory

In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, the distribution of zeros of the Riemann zeta function (and other L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...

s) is modelled 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.

## Gaussian ensembles

The most studied random matrix ensembles are the Gaussian ensembles.

The Gaussian unitary ensemble GUE(n) is described by the Gaussian measure
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces...

with density

on the space of n × n Hermitian matrices H = (Hij). Here ZGUE(n) = 2n/2 n2/2 is a normalisation 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
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

lacking time-reversal symmetry.

The Gaussian orthogonal ensemble GOE(n) is described by the Gaussian measure with density

on the space of n × n real symmetric matrices H = (Hij). Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry.

The Gaussian symplectic ensemble GSE(n) is described by the Gaussian measure with density

on the space of n × n quaternionic Hermitian matrices H = (Hij). Its distribution is invariant under conjugation by the symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...

, and it models Hamiltonians with time-reversal symmetry but no rotational symmetry.

The joint probability density
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

for the eigenvalues
Eigenvalue, eigenvector and eigenspace
The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

λ1,λ2,...,λn of GUE/GOE/GSE is given by

where β = 1 for GOE, β = 2 for GUE, and β = 4 for GSE; Zβ,n is a normalisation constant which can be explicitly computed, see Selberg integral. In the case of GUE (β = 2), the formula (1) describes a determinantal point process
Determinantal point process
In mathematics, a determinantal point process is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. Such processes arise as important tools in random matrix theory, combinatorics, and physics....

.

## Generalisations

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.

## 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.

### Global regime

In the global regime, one is interested in the distribution of linear statistics of the form Nf, H = n-1 tr f(H).

#### 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
Self-averaging
A self-averaging physical property of a disordered system is one that can be described by averaging over a sufficiently large sample. The concept was introduced by Ilya Mikhailovich Lifshitz.- Definition :...

. The cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

of the limiting measure is called the integrated density of states
Density of states
In solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...

and is denoted N(λ). If the integrated density of states is differentiable, its derivative is called the density of states
Density of states
In solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...

and is denoted ρ(λ).

The limit of the empirical spectral measure for Wigner matrices was described by Eugene Wigner, see Wigner's law. A more general theory was developed by Marchenko 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
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

.

#### Fluctuations

For the linear statistics Nf,H = n-1 ∑ f(λj), one is also interested in the fluctuations about ∫ f(λdN(λ). For many classes of random matrices, a central limit theorem of the form
is known, see et cet.

### Local regime

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.

#### Bulk statistics

Formally, fix λ0 in the interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....

of the support
Support (measure theory)
In mathematics, the support of a measure μ on a measurable topological space is a precise notion of where in the space X the measure "lives"...

of N(λ). Then consider the point process
Point process
In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces...

where λj are the eigenvalues of the random matrix.

The point process Ξ(λ0) captures the statistical properties of eigenvalues in the vicinity of λ0. For the Gaussian ensembles, the limit of Ξ(λ0) is known; thus, for GUE it is a determinantal point process
Determinantal point process
In mathematics, a determinantal point process is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. Such processes arise as important tools in random matrix theory, combinatorics, and physics....

with the kernel

(the sine kernel).

The universality principle postulates that the limit of Ξ(λ0) as n → ∞ should depend only on the symmetry class of the random matrix (and neither on the specific model of random matrices nor on λ0). This was rigorously proved for several models of random matrices: for invariant matrix ensembles, for Wigner matrices, et cet.

### Wishart matrices

Wishart matrices are n × n random matrices of the form H = X X*, where X is an n × n random matrix with independent entries, and X* is its conjugate matrix. 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
Vladimir Marchenko is a Ukrainian mathematician who specializes in mathematical physics, in particular in the analysis of the Sturm–Liouville operators. He introduced one of the approaches to the inverse problem for Sturm–Liouville operators...

and Leonid Pastur
Leonid Pastur
Leonid Andreevich Pastur is a mathematical physicist and theoretical physicist, known in particular for contributions to random matrix theory, the spectral theory of random Schrödinger operators, statistical mechanics, and solid state physics...

, see Marchenko–Pastur distribution
Marchenko–Pastur distribution
In random matrix theory, the Marchenko–Pastur distribution, or Marchenko–Pastur law, describes the asymptotic behavior of singular values of large rectangular random matrices...

.

## Guide to references

• Books on random matrix theory:
• Survey articles on random matrix theory:
• Historic works: