Wigner semicircle distribution

{\pi R^2} + \frac{\arcsin\!\left(\frac{x}{R}\right)}{\pi} | mean = $0$ | median = $0$ | mode = $0$ | variance = $\backslash frac\{R^2\}\{4\}$ | skewness = $0$ | kurtosis = $-1$ | entropy = | mgf = $2\backslash ,\backslash frac\{I\_1(R\; t)\}\{R\; t\}$ | char = $2\backslash ,\backslash frac\{J\_1(R\; t)\}\{R\; t\}$ }}

Wigner semicircle distribution. This continuous probability distribution is named for the physicist Eugene Wigner, who introduced it in the context of nuclear physics. It describes the limiting spectral density of large random matrices in the Gaussian unitary ensemble, a fundamental result in random matrix theory. The distribution's characteristic semicircular shape has made it a cornerstone in fields ranging from quantum chaos to telecommunications.

Definition

The probability density function for a radius $R\; >\; 0$ is defined on the interval $[-R,\; R]$. It is proportional to the function describing a semicircle, specifically $\backslash sqrt\{R^2\; -\; x^2\}$. The distribution is symmetric about zero, with its functional form emerging naturally from the eigenvalue statistics of certain matrix ensembles. This definition is intrinsically linked to the work of Hermann Weyl on group representations and the foundational studies of John von Neumann in operator theory. The normalization constant ensures the total probability integrates to one, a principle central to all probability theory.

Properties

The distribution is symmetric and unimodal, with its mean, median, and mode all located at zero. Its variance is determined solely by the radius parameter $R$. The characteristic function involves the Bessel function of the first kind, $J\_1$, while the moment-generating function involves the modified Bessel function $I\_1$. The distribution exhibits zero skewness and a negative excess kurtosis, indicating it is platykurtic relative to the normal distribution. These properties were extensively analyzed by researchers like Freeman Dyson and Madan Lal Mehta in their treatises on random matrices.

Related distributions

The Wigner semicircle law is the limiting case for the eigenvalue distribution of the Gaussian orthogonal ensemble and the Gaussian symplectic ensemble, with scaling factors differing by constants. It is intimately connected to the Marchenko–Pastur distribution, which governs singular values of certain random matrices. In the context of free probability, introduced by Dan Voiculescu, it plays the role analogous to the normal distribution, being the limiting law for sums of freely independent non-commutative random variables. This links it to the R-transform and the Stieltjes transformation in complex analysis.

Occurrence and applications

Its primary occurrence is in random matrix theory, where it describes the asymptotic eigenvalue density of large Hermitian matrices with independent entries, a result solidified by the work of László Erdős and others. Applications extend to quantum gravity models, like those studied by Edward Witten, and to the spectral analysis of large covariance matrices in multivariate statistics. It also appears in the study of quantum chromodynamics and in the performance analysis of large MIMO systems in wireless communications, a field advanced by institutions like Bell Labs.

Moments and cumulants

The moments of the distribution are given by the Catalan numbers for even orders, while all odd moments are zero due to symmetry. Specifically, the $2k$-th moment is $C\_k\; (R/2)^\{2k\}$, where $C\_k$ is the $k$-th Catalan number. The cumulants, except for the second (variance), are not as simply expressed in closed form but are foundational in the combinatorial framework of free probability developed by Roland Speicher. The moment sequence connects deeply to enumerative problems in combinatorics, such as non-crossing partitions counted in the work of Germain Kreweras.

Category:Continuous distributions Category:Random matrix theory