The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval the graph of whose probability density functionf is a semicircle of radius R centered at and then suitably normalized : for −R ≤ x ≤ R, and f = 0 if |x| > R. This distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity. It is a scaled beta distribution, more precisely, if Y is beta distributed with parameters α = β = 3/2, then X = 2RY – R has the above Wigner semicircle distribution. A higher-dimensional generalization is a parabolic distribution in three dimensional space, namely the marginal distribution function of a spherical distribution Note that R=1. While Wigner's semicircle distribution pertains to the distribution of eigenvalues, Wigner surmise deals with the probability density of the differences between consecutive eigenvalues.
In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory. Namely, in free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all partitions of a finite set in the theory of ordinary cumulants is replaced by the set of all noncrossing partitions of a finite set. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution.
Related distributions
Wigner (spherical) parabolic distribution
The parabolic probability distribution supported on the interval of radius R centered at : for −R ≤ x ≤ R, and f = 0 if |x| > R. Example. The joint distribution is Hence, the marginal PDF of the spherical distribution is such that R=1 The characteristic function of a spherical distribution becomes the pattern multiplication of the expected values of the distributions in X, Y and Z. The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals.
Wigner n-sphere distribution
The normalized N-sphere probability density function supported on the interval of radius 1 centered at : for −1 ≤ x ≤ 1, and f = 0 if |x| > 1. Example. The joint distribution is Hence, the marginal PDF distribution is such that R=1 The cumulative distribution function is such that R=1 and n >= -1 The characteristic function of the PDF is related to the beta distribution as shown below In terms of Bessel functions this is Raw moments of the PDF are Central moments are The corresponding probability moments are: Raw moments of the characteristic function are: For an even distribution the moments are Hence, the moments of the CF are Skew and Kurtosis can also be simplified in terms of Bessel functions. The entropy is calculated as The first 5 moments, such that R=1 are
N-sphere Wigner distribution with odd symmetry applied
The marginal PDF distribution with odd symmetry is such that R=1 Hence, the CF is expressed in terms of Struve functions "The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by"
Example (Normalized Received Signal Strength): quadrature terms
The normalized received signal strength is defined as and using standard quadrature terms Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining The expanded form of the Characteristic function of the received signal strength becomes
