Circular ensemble
Encyclopedia
In the theory of random matrices
, the circular ensembles are measures on spaces of unitary matrices introduced by Freeman Dyson
as modifications of the Gaussian matrix ensembles. The three main examples are the orthogonal circular ensemble (COE) on symmetric unitary matrices, the unitary circular ensemble (CUE) on unitary matrices, and the symplectic circular ensemble (CSE) on self dual unitary quaternionic matrices.
The distribution of the unitary circular ensemble CUE(n) is the Haar measure
on the unitary group
U(n). If U is a random element of CUE(n), then UTU is a random element of COE(n); if U is a random element of CUE(2n), then URU is a random element of CSE(n), where
The joint distribution of eigenvalues of the circular ensembles is given by the probability density function
where β=1 for COE, β=2 for CUE, and β=4 for CSE. The normalisation constant Zn,β is given by
Integrals of products of matrix coefficients in the unitary circular ensemble can be calculated using Weingarten function
s.
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...
, the circular ensembles are measures on spaces of unitary matrices introduced by Freeman Dyson
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...
as modifications of the Gaussian matrix ensembles. The three main examples are the orthogonal circular ensemble (COE) on symmetric unitary matrices, the unitary circular ensemble (CUE) on unitary matrices, and the symplectic circular ensemble (CSE) on self dual unitary quaternionic matrices.
The distribution of the unitary circular ensemble CUE(n) is the Haar measure
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
on the unitary group
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
U(n). If U is a random element of CUE(n), then UTU is a random element of COE(n); if U is a random element of CUE(2n), then URU is a random element of CSE(n), where
The joint distribution of eigenvalues of the circular ensembles is given by the probability density function
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...
where β=1 for COE, β=2 for CUE, and β=4 for CSE. The normalisation constant Zn,β is given by
Integrals of products of matrix coefficients in the unitary circular ensemble can be calculated using Weingarten function
Weingarten function
In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups...
s.