# Wigner semicircle distribution

The **Wigner semicircle distribution**, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−*R*, *R*] the graph of whose probability density function *f* is a semicircle of radius *R* centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse):

Probability density function | |||

Cumulative distribution function | |||

Parameters | radius (real) | ||
---|---|---|---|

Support | |||

CDF |
for | ||

Mean | |||

Median | |||

Mode | |||

Variance | |||

Skewness | |||

Ex. kurtosis | |||

Entropy | |||

MGF | |||

CF |

for −*R* ≤ *x* ≤ *R*, and *f*(*x*) = 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* = 2*RY* – *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 (parametric) distribution[1][2]

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.

## General properties

The Chebyshev polynomials of the second kind are orthogonal polynomials with respect to the Wigner semicircle distribution.

For positive integers *n*, the 2*n*-th moment of this distribution is

where *X* is any random variable with this distribution and *C*_{n} is the *n*th Catalan number

so that the moments are the Catalan numbers if *R* = 2. (Because of symmetry, all of the odd-order moments are zero.)

Making the substitution into the defining equation for the moment generating function it can be seen that:

which can be solved (see Abramowitz and Stegun §9.6.18) to yield:

where is the modified Bessel function. Similarly, the characteristic function is given by:[3][4]

where is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving is zero.)

In the limit of approaching zero, the Wigner semicircle distribution becomes a Dirac delta function.

## Relation to free probability

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

Parameters | radius (real) | ||
---|---|---|---|

Support | |||

CDF | |||

MGF | |||

CF |

The parabolic probability distribution supported on the interval [−*R*, *R*] of radius *R* centered at (0, 0):

for −*R* ≤ *x* ≤ *R*, and *f*(*x*) = 0 if *|x|* > *R*.

**Example.** The joint distribution is

Hence, the marginal PDF of the spherical (parametric) distribution is [1]

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 [−1, 1] of radius 1 centered at (0, 0):

,

for −1 ≤ *x* ≤ 1, and *f*(*x*) = 0 if *|x|* > 1.

**Example.** The joint distribution is

Hence, the marginal PDF distribution is [1]

such that R=1

The cumulative distribution function (CDF) is

such that R=1 and n >= -1

The characteristic function (CF) 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 (mean, variance, skew, kurtosis and excess-kurtosis) are:

Raw moments of the characteristic function are:

For an even distribution the moments are

Hence, the moments of the CF (provided N=1) are

Skew and Kurtosis can also be simplified in terms of Bessel functions.

The entropy is calculated as

The first 5 moments (n=-1 to 3), such that R=1 are

## N-sphere Wigner distribution with odd symmetry applied

The marginal PDF distribution with odd symmetry is [1]

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" [5]

## 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 [6]

## See also

- Wigner surmise
- The W.s.d. is the limit of the Kesten–McKay distributions, as the parameter
*d*tends to infinity. - In number-theoretic literature, the Wigner distribution is sometimes called the Sato–Tate distribution. See Sato–Tate conjecture.
- Marchenko–Pastur distribution or Free Poisson distribution

## References

- Buchanan, K.; Huff, G. H. (July 2011). "A comparison of geometrically bound random arrays in euclidean space".
*2011 IEEE International Symposium on Antennas and Propagation (APSURSI)*: 2008–2011. doi:10.1109/APS.2011.5996900. ISBN 978-1-4244-9563-4. - Buchanan, K.; Flores, C.; Wheeland, S.; Jensen, J.; Grayson, D.; Huff, G. (May 2017). "Transmit beamforming for radar applications using circularly tapered random arrays".
*2017 IEEE Radar Conference (RadarConf)*: 0112–0117. doi:10.1109/RADAR.2017.7944181. ISBN 978-1-4673-8823-8. - https://ieeexplore.ieee.org/document/7944181/
- https://oaktrust.library.tamu.edu/handle/1969.1/157918
- W., Weisstein, Eric. "Struve Function".
*mathworld.wolfram.com*. Retrieved 2017-07-28. - "Advanced Beamforming for Distributed and Multi-Beam Networks" (PDF).

- Milton Abramowitz and Irene A. Stegun, eds.
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.*New York: Dover, 1972.

## External links

- Eric W. Weisstein et al., Wigner's semicircle