WignerSemicircleDistribution
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WignerSemicircleDistribution
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represents a Wigner semicircle distribution with radius r centered at the origin.
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represents a Wigner semicircle distribution with radius r centered at a.
Details
- The probability density for value
in a Wigner semicircle distribution is proportional to 
for 
between 
and 
. - WignerSemicircleDistribution allows a to be any real number and r to be any positive real number.
- WignerSemicircleDistribution allows a and r to be any quantities of the same unit dimensions. »
- WignerSemicircleDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- WignerSemicircleDistribution[a,r] represents a continuous statistical distribution supported over the interval
that is parametrized by a real number a (the center of the distribution) and a positive real number r (its radius), which together determine the overall behavior of its probability density function (PDF). In general, the PDF of a Wigner semicircle distribution is shaped like an ellipse and hence is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the values of a and r. The one-parameter form WignerSemicircleDistribution[r] is equivalent to WignerSemicircleDistribution[0,r]. In general, the Wigner semicircle distribution may be referred to as Wigner's semicircle, the Sato–Tate distribution, or simply as the Wigner distribution. - The Wigner semicircle distribution is named for Hungarian-American scientist Eugene Wigner, whose 1955 treatise on infinite random matrices showed the importance of the distribution within the context of random matrix theory. The Wigner semicircle distribution is the statistical foundation of the so-called Wigner semicircle law, which states that the eigenvalues of a certain class of random matrices arising in quantum mechanics are Wigner distributed, Many of the applications of the distribution are related to phenomena in statistical quantum mechanics and various subfields of mathematics, including linear algebra, Fourier series, and spectral analysis.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Wigner distribution. Distributed[x,WignerSemicircleDistribution[a,r]], written more concisely as xWignerSemicircleDistribution[a,r], can be used to assert that a random variable x is distributed according to a Wigner distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions for Wigner distributions may be given using PDF[WignerSemicircleDistribution[a,r],x] and CDF[WignerSemicircleDistribution[a,r],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with a Wigner distribution, EstimatedDistribution to estimate a Wigner parametric distribution from given data, and FindDistributionParameters to fit data to a Wigner distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Wigner distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Wigner distribution.
- TransformedDistribution can be used to represent a transformed Wigner distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Wigner distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Wigner distributions.
- WignerSemicircleDistribution is related to a number of other distributions. It is generalized by PearsonDistribution, in the sense that the PDF of WignerSemicircleDistribution[r] is precisely the same as that of PearsonDistribution[1,1,0,-1,0,r] for r>0. Similarly, WignerSemicircleDistribution can be realized as a transformation (TransformedDistribution) of BetaDistribution, while PDF[WignerSemicircleDistribution[r],x] tends to DiracDelta[x] as r→0. WignerSemicircleDistribution is also related to VonMisesDistribution and BetaPrimeDistribution.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-ux2bof
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-pf2r7l
Out[2]=2
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-jfj5an
Out[2]=2
In[4]:=4
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-l4o8g8
Out[4]=4
Cumulative distribution function:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-zj314a
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-d3ajok
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-vwokpi
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-u83yqp
Out[4]=4
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-vbfs2x
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-jecqsf
Out[2]=2
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-vkfoux
Out[1]=1
Scope (7)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a Wigner semicircle distribution:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-qhtk5j
Compare its histogram to the PDF:
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-03mwaz
Out[2]=2
Distribution parameters estimation:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-45b7g2
Estimate the distribution parameters from sample data:
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-epi747
Out[2]=2
Compare the density histogram of the sample with the PDF of the estimated distribution:
In[3]:=3
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-f8ui5o
Out[3]=3
Skewness and kurtosis are constant:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-mncyr8
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-v7qg0f
Out[2]=2
Different moments with closed forms as functions of parameters:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-js043h
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-rx074o
Out[2]=2
Closed form for symbolic order:
In[3]:=3
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-m9swpo
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-pknsqa
Out[4]=4
Closed form for symbolic order:
In[5]:=5
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-4052gd
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-zg9ct4
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-9gzmth
Out[7]=7
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-w9ogck
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-ffgpq1
Out[2]=2
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-qccu4o
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-6vqfb2
Out[2]=2
Consistent use of Quantity in parameters yields QuantityDistribution:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-bf9qsq
Out[1]=1
Find the quartiles of the spectrum:
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-0vdsl
Out[2]=2
Applications (1)Sample problems that can be solved with this function
Illustrate universality conjecture of the Wigner semicircle law asserting that spectral density of large symmetric matrices, formed from independent identically distributed random variables with zero mean and finite variance, converges to PDF of WignerSemicircleDistribution:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-gmd7xd
Sample the spectrum of matrices with elements drawn from Student 
distribution:
In[22]:=22
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-b7zdy
Use method of moments to fit WignerSemicircleDistribution to the sample:
In[28]:=28
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-lvitr
Out[28]=28
In[29]:=29
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-np318p
Out[29]=29
The conjecture does not hold for distributions whose variance is not finite:
In[17]:=17
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-n8wwoo
In[21]:=21
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-hd0gcg
Out[21]=21
In[19]:=19
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-d9n6sx
Out[19]=19
Properties & Relations (4)Properties of the function, and connections to other functions
The WignerSemicircleDistribution is closed under translation and scaling by a positive factor:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-p5q1zr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-z9k4te
Out[2]=2
Relationships to other distributions:
Wigner semicircle distribution is a special case of type 1 PearsonDistribution:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-0y5as3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-smoa0u
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-ylwno8
Out[3]=3
Wigner semicircle distribution is a transformation of special BetaDistribution:
In[1]:=1
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-zo9b0n
In[2]:=2
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-djj5oc
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-ng3g1m
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0hu9k4akk8tp6dimstmlqy-s0lugl
Out[4]=4
Wolfram Research (2010), WignerSemicircleDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html (updated 2016).
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Wolfram Research (2010), WignerSemicircleDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html (updated 2016).Text
Wolfram Research (2010), WignerSemicircleDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html (updated 2016).
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Wolfram Research (2010), WignerSemicircleDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html (updated 2016).CMS
Wolfram Language. 2010. "WignerSemicircleDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html.
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Wolfram Language. 2010. "WignerSemicircleDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html.APA
Wolfram Language. (2010). WignerSemicircleDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html
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Wolfram Language. (2010). WignerSemicircleDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.htmlBibTeX
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@misc{reference.wolfram_2025_wignersemicircledistribution, author="Wolfram Research", title="{WignerSemicircleDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html}", note=[Accessed: 25-March-2025
]}BibLaTeX
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@online{reference.wolfram_2025_wignersemicircledistribution, organization={Wolfram Research}, title={WignerSemicircleDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html}, note=[Accessed: 25-March-2025
]}