In
probability theoryProbability 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 physicsMathematical 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
matrixIn 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 variableIn 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 systemIn 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 conductivityIn 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
latticeIn 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 physicsNuclear 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 physicsSolid-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
HamiltoniansIn 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 chaosQuantum 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 gravityQuantum 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 statisticsMultivariate 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 WishartJohn 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 matricesIn 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 analysisNumerical 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 NeumannJohn 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 multiplicationIn 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...
. See also for more recent results.
Number theory
In
number theoryNumber 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-functionThe 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 measureIn 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) = 2
n/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
HamiltoniansIn 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 groupIn 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 densityIn 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
eigenvaluesThe 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 processIn 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-averagingA 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 functionIn 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 statesIn 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 statesIn 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 theoryIn 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(
λ)
dN(
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
interiorIn 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
supportIn 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 processIn 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 processIn 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 MarchenkoVladimir 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 PasturLeonid 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 distributionIn 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: