Dyson index β
E443154
The Dyson index β is a parameter in random matrix theory that classifies ensembles by their underlying symmetry, typically taking values 1, 2, or 4 for orthogonal, unitary, and symplectic ensembles, respectively.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Dyson symmetry class parameter
ⓘ
parameter in random matrix theory ⓘ symmetry index ⓘ |
| appearsIn |
Vandermonde determinant exponent
ⓘ
Wigner–Dyson statistics NERFINISHED ⓘ joint eigenvalue distribution ⓘ |
| canTake | positive real values in generalized β-ensembles ⓘ |
| classifies |
Gaussian ensembles
NERFINISHED
ⓘ
circular ensembles ⓘ random matrix ensembles ⓘ |
| classifiesBy | underlying symmetry ⓘ |
| controls | level repulsion exponent ⓘ |
| determines |
short-range spectral correlations
ⓘ
spacing distribution between eigenvalues ⓘ |
| field |
mathematical physics
ⓘ
probability theory ⓘ spectral theory ⓘ |
| generalizationOf | Wigner’s symmetry classification NERFINISHED ⓘ |
| introducedInContextOf | Dyson’s threefold way NERFINISHED ⓘ |
| mathematicalRole | exponent in eigenvalue interaction term ⓘ |
| namedAfter | Freeman Dyson NERFINISHED ⓘ |
| relatedTo |
spin-rotation symmetry
ⓘ
time-reversal symmetry ⓘ universality classes in random matrix theory ⓘ |
| takesTypicalValue |
1
ⓘ
2 ⓘ 4 ⓘ |
| usedIn |
mesoscopic physics
ⓘ
nuclear physics ⓘ number theory ⓘ quantum chaos ⓘ random matrix theory NERFINISHED ⓘ |
| usedToLabel |
Gaussian β-ensembles
NERFINISHED
ⓘ
Jacobi β-ensembles NERFINISHED ⓘ Laguerre β-ensembles NERFINISHED ⓘ |
| value1AssociatedWith |
Gaussian orthogonal ensemble
NERFINISHED
ⓘ
circular orthogonal ensemble ⓘ |
| value1CorrespondsTo | orthogonal symmetry ⓘ |
| value2AssociatedWith |
Gaussian unitary ensemble
NERFINISHED
ⓘ
circular unitary ensemble ⓘ |
| value2CorrespondsTo | unitary symmetry ⓘ |
| value4AssociatedWith |
Gaussian symplectic ensemble
NERFINISHED
ⓘ
circular symplectic ensemble ⓘ |
| value4CorrespondsTo | symplectic symmetry ⓘ |
| βEquals1Implies | presence of time-reversal symmetry and integer spin ⓘ |
| βEquals2Implies | absence of time-reversal symmetry ⓘ |
| βEquals4Implies | presence of time-reversal symmetry with half-integer spin ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.