Concept
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): 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. 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 where X is any random variable with this distribution and Cn is the nth 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: 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. In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory.
