Wigner matrices
E898468
Wigner matrices are large random symmetric (or Hermitian) matrices with independent, identically distributed entries (up to symmetry) that serve as a fundamental model in random matrix theory for studying eigenvalue statistics and universal spectral behavior.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical object
ⓘ
random matrix ensemble ⓘ |
| appliesTo |
complex disordered quantum systems
ⓘ
models of heavy nuclei energy levels ⓘ |
| entryDistributionCondition |
diagonal entries are independent and identically distributed
ⓘ
entries have finite variance ⓘ off-diagonal entries are independent and identically distributed ⓘ off-diagonal entries are independent of diagonal entries ⓘ |
| field |
mathematical physics
ⓘ
probability theory ⓘ random matrix theory ⓘ |
| hasProperty |
centered entries (often mean zero)
ⓘ
complex Hermitian in the complex case ⓘ eigenvalues are real ⓘ identically distributed diagonal entries (possibly different law from off-diagonal) ⓘ identically distributed off-diagonal entries ⓘ independent entries up to symmetry ⓘ large dimension ⓘ real symmetric in the real case ⓘ symmetric or Hermitian ⓘ variance-normalized entries ⓘ |
| hasVariant |
Gaussian Orthogonal Ensemble
NERFINISHED
ⓘ
Gaussian Unitary Ensemble NERFINISHED ⓘ complex Hermitian Wigner matrices ⓘ real symmetric Wigner matrices ⓘ |
| introducedBy | Eugene Wigner NERFINISHED ⓘ |
| introducedInContext | statistical theory of energy levels of complex quantum systems ⓘ |
| namedAfter | Eugene Wigner NERFINISHED ⓘ |
| relatedTo |
Dyson Brownian motion
NERFINISHED
ⓘ
free probability theory ⓘ level repulsion ⓘ semicircle distribution ⓘ universality conjectures ⓘ |
| satisfies |
Wigner semicircle law for empirical spectral distribution
ⓘ
concentration of spectral norm around 2 in the normalized case ⓘ universality of local eigenvalue statistics under mild moment conditions ⓘ |
| scalingConvention | entries often scaled by 1/sqrt(n) ⓘ |
| typicalResult |
empirical spectral measure converges almost surely to semicircle law
ⓘ
fluctuations of largest eigenvalue often follow Tracy–Widom distribution ⓘ largest eigenvalue converges to 2 in the normalized case ⓘ local eigenvalue statistics in bulk match GOE or GUE statistics ⓘ |
| usedFor |
modeling energy levels of complex quantum systems
ⓘ
proving universality results in random matrix theory ⓘ studying delocalization of eigenvectors ⓘ studying eigenvalue statistics ⓘ studying global eigenvalue distribution ⓘ studying local eigenvalue spacing ⓘ studying spectral norms of random matrices ⓘ studying universal spectral behavior ⓘ |
Referenced by (1)
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