Semicircle Distribution

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The semicircle distribution (also called Wigner’s semicircle distribution) is a continuous probability distribution shaped like a scaled semicircle. It is named after physicist Eugene Wigner (1902-1995).

wigner's semicircle distribution CC0
Wigner semicircle distribution is centered at the origin (0, 0) with radius R > 0 on the interval [-R, R].

The probability density function (PDF) of the semicircle distribution is:

PDF for Wigner semicircle distribution

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

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):

CDF for the semicircle distribution.

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.

Use 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].


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


[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:

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

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