In random matrix theory, the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled from the gaussian distribution. They are among the most-commonly studied matrix ensembles, fundamental to both mathematics and physics. The three main examples are the Gaussian orthogonal (GOE), unitary (GUE), and symplectic (GSE) ensembles. These are classified by the Dyson index β, which takes values 1, 2, and 4 respectively, counting the number of real components per matrix element (1 for real elements, 2 for complex elements, 4 for quaternions). The index can be extended to take any real positive value.

The gaussian ensembles are also called the Wigner ensembles, or the Hermite ensembles.

Definitions

Conventions

There are many conventions for defining the Gaussian ensembles. In this article, we specify exactly one of them.

In all definitions, the Gaussian ensemble have zero expectation.

• : a positive real number. Called the Dyson index are special.

• : the side-length of a matrix. Always a positive integer.

• stands for "Wigner".

: the adjoint (self-adjoint is sampled from a gaussian ensemble.

• is its transpose.

• is its conjugate transpose.

• : the eigenvalues of the matrix, which are all real, since the matrices are always assumed to be self-adjoint.

• .

.

• .

• For a quaternion.

• : the partition function.

When referring to the main reference works, it is necessary to translate the formulas from them, since each convention leads to different constant scaling factors for the formulas.There are equivalent definitions for the GβE(N) ensembles, given below.

By sampling

cases, the GβE(N) ensemble is defined by how it is sampled:

, such that all its entries are IID sampled from the corresponding standard normal distribution.

• .

• .

• .

• .

By density

.

The Gaussian orthogonal ensemble.

as the degrees of freedom.

The Gaussian unitary ensemble.

as the degrees of freedom.

The Gaussian symplectic ensemble.

as the degrees of freedom.

By invariance

cases, the GβE(N) ensemble is uniquely characterized (up to affine transform) by its symmetries, or invariance under appropriate transformations.

symmetric matrices satisfying the following properties:

• Invariance under orthogonal transformation orthogonal matrix.

• Independence are independently distributed.

Hermitian matrices satisfying the following properties:

• Invariance under unitary transformation unitary matrix.

• are independently distributed.

self-adjoint quaternionic matrices satisfying the following properties:

• Invariance under symplectic transformation symplectic matrix.

• are independently distributed.

, we recover the GOE, GUE, GSE..

More succinctly stated, each of GOE, GUE, GSE is uniquely specified by invariance, independence, the mean, and the variance.

By spectral distribution

, where

• is a diagonal real matrix with its entries sampled according to the spectral density, defined below;

• is an orthogonal/unitary/symplectic matrix sampled uniformly, that is, from the normalized Haar measure of the orthogonal/unitary/symplectic group.

In this way, the GβE(N) ensemble may be defined after the spectral density is defined first, so that any method to motivate the spectral density then motivates the GβE(N) ensemble, and vice versa.

By maximal entropy

real/complex/quaternionic symmetric/orthogonal/symplectic matrices that maximizes entropy.

Spectral density

the joint density of Gβ is the Vandermonde determinant is explicitly evaluated as a Selberg integral: is the Euler Gamma function, where we have a superfactorial

Determinantal point process

is the probabilist's Hermite polynomial. These are the wavefunction states of the quantum harmonic oscillator.

The spectrum of GUE(N) is a determinantal point process, and by the Christoffel–Darboux formula: can also be written as a quaternionic determinantal point process involving skew-orthogonal polynomials.

Tridiagonalization

from the GβE(N) ensemble, we can perform a Householder transformation tridiagonalization on it to obtain a tridiagonal matrix is chi-distributed case was first noted in 1984, and the general case was noted in 2002. Like how the Laplace differential operator can be discretized to the Laplacian matrix, this tridiagonal form of the gaussian ensemble allows a reinterpretation of the gaussian ensembles as an ensemble over not matrices, but over differential operators, specifically, a "stochastic Airy operator". This leads more generally to the study of random matrices as stochastic operators.

, by using the Sturm sequences. cases, leading to the gaussian beta ensembles, and "anti-symmetric" gaussian beta ensembles.

is a Householder matrix.

Global law

The Wigner semicircle law converges in distribution to the Wigner semicircle distribution with radius 2.

The requirement that the matrix ensemble to be a gaussian ensemble is too strong for the Wigner semicircle law. Indeed, the theorem applies generally for much more generic matrix ensembles.

As Coulomb gas

can be written as a Gibbs measurewith the energy function (also called the Hamiltonian. This can be interpreted physically as a Boltzmann distribution identical unit electric charges constrained to move on the real line, repelling each other via the two-dimensional Coulomb potential. This is the Coulomb gas model for the eigenvalues.

, where the mean-field, which gives the limiting eigenvalue density.

, which can be found by solving an integral equation. This indicates that the Wigner semicircle distribution is the equilibrium distribution.

to second order produce the sine kernel in the bulk and the Airy kernel at the soft edge after proper rescaling.

Extreme value

The largest eigenvalue for GβE(N) follows the Tracy–Widom distribution after proper translation and scaling. upper left corner of the tridiagonal matrix form.

Level spacings

Moments

For GOE(N), its moment generating function is the Frobenius norm.

Rotationally symmetric ensemble

The GUE(N) ensemble can be understood as a particularly representative member of the general family of rotationally symmetric ensembles over Hermitian matrices. The general theory of rotationally symmetric ensembles has allowed researchers to prove some universality results.

Physics

The Wigner surmise

The Gaussian ensemble was first motivated in theoretical physics. In the 1940s, Eugene Wigner studied the irregular spacings of slow-neutron resonances in heavy nuclei. Working with the few dozen levels then available, he noticed a pronounced repulsion between neighbouring lines.

In 1951, he modelled the Hamiltonian of a compound-nucleus in a minimal way., which matched well with the data. He disseminated his guess ("the Wigner surmise") during a conference on Neutron Physics by Time-of-Flight in 1956:Freeman Dyson stated the project as a statistical theory of nuclear energy levels, to be contrasted with precise calculations based on an analytic model of the nucleus. He argued that a statistical theory is necessary, because the energy levels then measured were on the order of millions, and for such a high order, precise calculations was simply impossible. The idea was different from the then-understood form of statistical mechanics, for instead of having a system with precisely stated dynamical laws, with too many particles interacting under it, thus the particles need to be treated statistically, he would model the dynamical laws themselves as unknown, and thus treated statistically.

Threefold Way

In 1962, Dyson proposed the "Threefold Way" to motivate the three ensembles, by showing that in 3 fields (group representation, quantum mechanics, random matrix theory), there is a 3-fold disjunction, which he traced back to the Frobenius theorem stating that there are only 3 real division algebras: the real, the complex, and the quaternionic. A random matrix representing a Hamiltonian can be classified by an anti-unitary that describes time-reversal symmetry, and the corresponding gaussian ensemble can then be motivated as a maximal entropy distribution, as described previously. must be real symmetric. This typically occurs in systems with no magnetic field and either spinless particles or integer spin particles with negligible spin–orbit interaction. This occurs in level spacing distribution in nuclear compound states, the original motivation for Wigner.

is only required to be Hermitian. Time-reversal symmetry can be broken by a homogeneous magnetic field, random magnetic fluxes, or spin-selective lasers. In these cases, the off-diagonal matrix elements acquire independent complex phases.

• Chaotic microwave cavities with a ferrite: Adding a strong axial magnetic field causes the level statistics to transition continuously from GOE to GUE, which was a confirmation of the BGS conjecture.

• Quantum Hall effect: The physics of quantum Hall edge states and Landau levels is modelled by the GUE due to the strong perpendicular magnetic field breaking time-reversal symmetry.

• Anderson localization in 3-D: Applying an Aharonov–Bohm flux can drive a system's statistics from GOE to GUE at a disorder-induced metal-insulator transition.

, then this is a consequence of Kramers' theorem for systems with half-integer spin and significant spin–orbit interaction. The resulting Hamiltonians are naturally described by quaternion-Hermitian matrices. It has been observed in Kramers doublet and many quantum chaotic systems. It is also possible to construct such a system without spin.

References

External links