The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution defined on the domain [−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 distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise.

For positive integers n, the 2n-th moment of this distribution is In the typical special case that R = 2, this sequence coincides with the Catalan numbers 1, 2, 5, 14, etc.

[1] As can be calculated using the residue theorem, the Stieltjes transform of the Wigner distribution is given by for complex numbers z with positive imaginary part, where the complex square root is taken to have positive imaginary part.

[5] In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory.