The semicircle distribution (also called Wigner’s semicircle distribution) is a continuous probability distribution shaped like a scaled semicircle. It is centered at the origin (0, 0) with radius R > 0 on the interval [-R, R].
The probability density function of the semicircle distribution is:
A variant is the power semicircle distribution PS(θ, R), which has PDF:
fθ(x, R) = c′θ, R(f0(x; R))2θ + 1) = cθ, R (R2 – x2)θ + ½ 1|x| ≤ , R
Where R is the range parameter and θ is the shape parameter.
Uses of the Semicircle Distribution
The semicircle distribution plays an important role in many areas of mathematics, including applied mathematics. For example, physicist E.P, 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 . The semicircle law also appears in physics, in a quantum Brownian motion on the free boson Fock space . The distribution is also the limiting distribution of a Markov chain of Young diagrams  and is the limiting distribution in the free version of the central limit theorem .
Graph of Wigner’s semicircle distribution created with Desmos.
 Benaych-Georges, F. & Knowles, A. Lectures on the local semicircle law for Wigner matrices.
 Hashimoto, Y. DEFORMATIONS OF ITHE SEMICIRCLE LAW DERIVED FROM RANDOM WALKS ON FREE GROUPS.
 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
 Barndorff-Nielsen, O. & Thorbjørnsen, S. Levy laws in free probability.
Stephanie Glen. "Semicircle Distribution" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/semicircle-distribution/
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