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. It is also a scaled beta distribution: if Y is beta-distributed with parameters α = β = 3/2, then X = 2RY – R has the Wigner semicircle distribution. The distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity. The distribution of the spacing 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 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. 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. Related distributions Wigner (spherical) parabolic distribution The parabolic probability distribution supported on the interval [−R,