The semicircular distribution of variance 
is the probability
distribution on the real line with density
for 
,
and zero otherwise. Its mean is zero and its variance
is 
.
It is the limiting distribution in Wigner's
semicircle law. A variance-one freely
independent family with this distribution supplies the noise in nonmicrostates
free entropy.
See also
Free Entropy,
Free Fisher Information,
Free Independence,
Wigner's Semicircle Law
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References
Voiculescu, D. "Free Entropy." Bull. London
Math. Soc. 34, 257-278, 2002. https://doi.org/10.1112/S0024609301008992.
Cite this as:
Weisstein, Eric W. "Semicircular Distribution."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SemicircularDistribution.html
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