Wigner semicircle distribution

Wigner semicircle

Probability density function
Cumulative distribution function
Parameters	$R>0\!$ radius (real)
Support	$x\in [-R;+R]\!$
PDF	${\frac {2}{\pi R^{2}}}\,{\sqrt {R^{2}-x^{2}}}\!$
CDF	${\frac {1}{2}}+{\frac {x{\sqrt {R^{2}-x^{2}}}}{\pi R^{2}}}+{\frac {\arcsin \!\left({\frac {x}{R}}\right)}{\pi }}\!$ for $-R\leq x\leq R$
Mean	$0\,$
Median	$0\,$
Mode	$0\,$
Variance	${\frac {R^{2}}{4}}\!$
Skewness	$0\,$
Excess kurtosis	$-1\,$
Entropy	$\ln(\pi R)-{\frac {1}{2}}\,$
MGF	$2\,{\frac {I_{1}(R\,t)}{R\,t}}$
CF	$2\,{\frac {J_{1}(R\,t)}{R\,t}}$

The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):

$f(x)={2 \over \pi R^{2}}{\sqrt {R^{2}-x^{2}\,}}\,$

for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. The parameter R is commonly referred to as the "radius" parameter of the distribution.

The Wigner distribution also coincides with a scaled beta distribution. That is, if Y is a beta-distributed random variable with parameters α = β = 3/2, then the random variable X = 2RY – R exhibits a Wigner semicircle distribution with radius R.

The distribution arises as the limiting distribution of the eigenvalues of many random symmetric matrices, that is, as the dimensions of the random matrix approach infinity. The distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise.

General properties

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

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

$E(X^{2n})=\left({R \over 2}\right)^{2n}C_{n}\,$

where X is any random variable with this distribution and C_n is the nth Catalan number

$C_{n}={1 \over n+1}{2n \choose n},\,$

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 $x=R\cos(\theta )$ into the defining equation for the moment generating function it can be seen that:

$M(t)={\frac {2}{\pi }}\int _{0}^{\pi }e^{Rt\cos(\theta )}\sin ^{2}(\theta )\,d\theta$

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

$M(t)=2\,{\frac {I_{1}(Rt)}{Rt}}$

where $I_{1}(z)$ is the modified Bessel function. Similarly, the characteristic function is given by:

$\varphi (t)=2\,{\frac {J_{1}(Rt)}{Rt}}$

where $J_{1}(z)$ is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving $\sin(Rt\cos(\theta ))$ is zero.)

In the limit of $R$ 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.

See also

• Wigner surmise
• The Wigner semicircle distribution 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

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.
• Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer (2010). An introduction to random matrices. Cambridge Studies in Advanced Mathematics. Vol. 118. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511801334. ISBN 978-0-521-19452-5. MR 2670897. Zbl 1184.15023.
• Bai, Zhidong; Silverstein, Jack W. (2010). Spectral analysis of large dimensional random matrices. Springer Series in Statistics (Second edition of 2006 original ed.). New York: Springer. doi:10.1007/978-1-4419-0661-8. ISBN 978-1-4419-0660-1. MR 2567175. Zbl 1301.60002.
• Wigner, Eugene P. "Characteristic vectors of bordered matrices with infinite dimensions". Annals of Mathematics. Second Series. 62 (3): 548–564. doi:10.2307/1970079. MR 0077805. Zbl 0067.08403.

External links

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