Gaussian symplectic ensemble
E444351
Gaussian ensemble
ensemble in random matrix theory
probability distribution on matrices
random matrix ensemble
The Gaussian symplectic ensemble is a random matrix ensemble of self-dual quaternionic Hermitian matrices used in random matrix theory to model systems with time-reversal symmetry and strong spin–orbit coupling.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Gaussian Symplectic Ensemble | 1 |
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Gaussian ensemble
ⓘ
ensemble in random matrix theory ⓘ probability distribution on matrices ⓘ random matrix ensemble ⓘ |
| abbreviation | GSE ⓘ |
| belongsToField |
mathematical physics
ⓘ
quantum chaos ⓘ random matrix theory ⓘ |
| contrastedWith |
Gaussian orthogonal ensemble (β = 1)
NERFINISHED
ⓘ
Gaussian unitary ensemble (β = 2) NERFINISHED ⓘ |
| hasAlternativeDescription | ensemble of complex 2N×2N Hermitian matrices with symplectic symmetry constraints ⓘ |
| hasApplicationsIn |
condensed matter physics
ⓘ
nuclear physics ⓘ quantum information theory ⓘ quantum transport ⓘ |
| hasCorrelationFunctions | expressible via quaternion determinants ⓘ |
| hasDysonClassLabel | β = 4 ⓘ |
| hasDysonIndex | 4 ⓘ |
| hasEdgeStatistics | Tracy–Widom distribution of type β = 4 for largest eigenvalue NERFINISHED ⓘ |
| hasEigenvalueRepulsionExponent | β = 4 ⓘ |
| hasEigenvalues | real ⓘ |
| hasGeneralization | non-Gaussian symplectic ensembles ⓘ |
| hasInvariance | invariant under conjugation by compact symplectic group ⓘ |
| hasJointEigenvalueDensity | proportional to exp(- (β/2) Σ λ_i^2 ) ∏_{i<j} |λ_i - λ_j|^β with β = 4 ⓘ |
| hasLevelRepulsion | P(s) ~ s^4 for small spacing s ⓘ |
| hasLevelSpacingStatistics | Wigner–Dyson distribution with β = 4 ⓘ |
| hasMatrixElementsDistribution | Gaussian in quaternion components ⓘ |
| hasMatrixSizeParameter | N ⓘ |
| hasMatrixType | self-dual quaternionic Hermitian matrices ⓘ |
| hasParameter | variance of matrix elements ⓘ |
| hasProbabilityDensityOnMatrices | proportional to exp(- (β/2) Tr H^2 ) with β = 4 ⓘ |
| hasSpectralMeasure | converges to Wigner semicircle law as N → ∞ ⓘ |
| hasSymmetryClass | symplectic symmetry ⓘ |
| hasSymmetryGroup | compact symplectic group Sp(N) ⓘ |
| hasTimeReversalSymmetry | true ⓘ |
| hasTypicalSymmetryClassLabel | class AII in Altland–Zirnbauer classification NERFINISHED ⓘ |
| hasUniversalityProperty | local eigenvalue statistics are universal in the bulk ⓘ |
| introducedBy | Freeman Dyson NERFINISHED ⓘ |
| introducedInContextOf | Dyson threefold way classification NERFINISHED ⓘ |
| isCompanionOf |
Gaussian orthogonal ensemble
NERFINISHED
ⓘ
Gaussian unitary ensemble NERFINISHED ⓘ |
| isOneOfDysonThreefoldWay | yes ⓘ |
| isPartOf | Wigner–Dyson ensembles NERFINISHED ⓘ |
| modelsSystemsWith | time-reversal symmetry and strong spin–orbit coupling ⓘ |
| relatedTo |
chiral Gaussian symplectic ensemble
ⓘ
symplectic Lie algebra ⓘ |
| usedToModel |
energy level statistics in complex nuclei
ⓘ
mesoscopic conductors with spin–orbit interaction ⓘ quantum systems with half-integer spin and time-reversal symmetry ⓘ systems with strong spin–orbit coupling ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gaussian Symplectic Ensemble