Probability Distributions > Semicircle distribution

## Wigner’s Semicircle Distribution

The semicircle distribution, also known as Wigner’s semicircle distribution, is a continuous probability phenomenon whose shape could be likened to the classic half-moon. Centered around origin (0, 0), this distribution stretches outwards in either direction with an adjustable radius of R > 0 over the interval [-R , R]. It is named after physicist Eugene Wigner (1902-1995).

The probability density function of the semicircle distribution is:

A variant is the **power semicircle distribution** PS(θ, R), which has PDF:

f_{θ}(x, R) = c′_{θ, R}(f_{0}(x; R))^{2θ + 1)} = c_{θ, R} (R^{2} – x^{2})^{θ + ½} 1_{|x| ≤ , R}

Where R is the range parameter and θ is the shape parameter. The range parameter determines the width of the distribution. It is a measure of the distance from the origin within which the distribution is concentrated. A larger range parameter means that the distribution is more spread out, while a smaller range parameter means that the distribution is more concentrated.

## Other semicircle distribution properties

Cumulative distribution function (CDF):

- Mean = Median = Mode = 0
- Variance = R
^{2}/4 - Skewness = 0
- Excess kurtosis = -1

In free probability theory (the study of *non-commutative random variables*), this distribution is equivalent to the normal distribution in classical probability theory. It is also a scaled beta distribution.

## Uses of the Semicircle Distribution

The semicircle distribution plays an important role in many areas of mathematics, including applied mathematics. For example, physicist Eugene Wigner showed it is the asymptotic spectral measure of Wigner ensembles of random matrices; the local semicircle law states that the eigenvalue distribution of a Wigner matrix is close to Wigner’s semicircle distribution [1]. The semicircle law also appears in physics, in a quantum Brownian motion on the free boson Fock space [2]. The distribution is also the limiting distribution of a Markov chain of Young diagrams [3] and is the limiting distribution in the free version of the central limit theorem [4].

## References

Graph of Wigner’s semicircle distribution created with Desmos.

[1] Benaych-Georges, F. & Knowles, A. Lectures on the local semicircle law for Wigner matrices.

[2] Hashimoto, Y. DEFORMATIONS OF ITHE SEMICIRCLE LAW DERIVED FROM RANDOM WALKS ON FREE GROUPS.

[3] Arizmendi, O. & Perez-Abreu, V. (2010). ON THE NON-CLASSICAL INFINITE DIVISIBILITY OF POWER SEMICIRCLE DISTRIBUTIONS. Communications on Stochastic Analysis. Vol. 4, No. 2. 161-178. Retrieved December 30, 2021 from: http://personal.cimat.mx:8181/~pabreu/4-2-02%5B221%5D.pdf

[4] Barndorff-Nielsen, O. & Thorbjørnsen, S. Levy laws in free probability.