page_title: Wigner semicircle distribution
section_title: Relation to free probability
hierarchical_section_title: Wigner semicircle distribution / Relation to free probability
caption_attribution_description: English: Characteristic Mode
context_page_description: The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at and then suitably normalized:
  for −R ≤ x ≤ R, and f(x) = 0 if |x| > R.
  This distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity.
  It is a scaled beta distribution, more precisely, if Y is beta distributed with parameters α = β = 3/2, then X = 2RY – R has the above Wigner semicircle distribution.
  A higher-dimensional generalization is a parabolic distribution in three dimensional space, namely the marginal distribution function of a spherical distribution
  Note that R=1.
  While Wigner's semicircle distribution pertains to the distribution of eigenvalues, Wigner surmise deals with the probability density of the differences between consecutive eigenvalues.
context_section_description: 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.
caption: Characteristic function spherical distribution
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