Wigner semicircle law
E898462
The Wigner semicircle law is a fundamental result in random matrix theory that describes how the eigenvalues of large random symmetric (or Hermitian) matrices are distributed according to a characteristic semicircular density.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
limit theorem
ⓘ
probability law ⓘ theorem in random matrix theory ⓘ |
| appliesTo |
large random Hermitian matrices
ⓘ
large random symmetric matrices ⓘ |
| assumes |
finite variance entries
ⓘ
identically distributed off-diagonal entries ⓘ independent matrix entries up to symmetry ⓘ mean-zero entries ⓘ |
| concernsLimit | matrix size n → ∞ ⓘ |
| concernsObject | Wigner matrices NERFINISHED ⓘ |
| concernsQuantity |
eigenvalue density
ⓘ
spectral measure ⓘ |
| describes |
empirical spectral distribution of eigenvalues
ⓘ
limiting eigenvalue distribution of random matrices ⓘ |
| field |
mathematical physics
ⓘ
probability theory ⓘ random matrix theory ⓘ |
| hasAlternativeName |
Wigner’s semicircle distribution
NERFINISHED
ⓘ
semicircular law NERFINISHED ⓘ |
| hasConvergenceType |
almost sure convergence
GENERATED
ⓘ
weak convergence of measures GENERATED ⓘ |
| hasDensityFormula | (1/(2πσ²))·sqrt(4σ² - x²) on [-2σ,2σ] ⓘ |
| hasDensityShape | semicircle ⓘ |
| hasGeneralization | free central limit theorem NERFINISHED ⓘ |
| hasParameter | variance parameter σ² ⓘ |
| hasSupport | compact interval ⓘ |
| hasSymmetry | symmetric density around 0 ⓘ |
| hasTypicalSupportForm | [-2σ, 2σ] ⓘ |
| holdsFor |
Gaussian Orthogonal Ensemble
NERFINISHED
ⓘ
Gaussian Symplectic Ensemble NERFINISHED ⓘ Gaussian Unitary Ensemble NERFINISHED ⓘ |
| implies | convergence of empirical spectral distribution ⓘ |
| introducedBy | Eugene Wigner NERFINISHED ⓘ |
| isAnalogOf | central limit theorem for eigenvalues ⓘ |
| isSpecialCaseOf | universality phenomena in random matrices ⓘ |
| mathematicalDomain |
matrix analysis
ⓘ
spectral theory ⓘ |
| namedAfter | Eugene Wigner NERFINISHED ⓘ |
| relatedTo |
Marchenko–Pastur law
NERFINISHED
ⓘ
circular law ⓘ free probability theory ⓘ |
| usedIn |
nuclear physics
ⓘ
number theory ⓘ quantum chaos ⓘ statistics of large data matrices ⓘ theory of disordered systems ⓘ |
| yearIntroducedApprox | 1950s ⓘ |
Referenced by (1)
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