Gaussian orthogonal ensemble
E443153
The Gaussian orthogonal ensemble is a fundamental random matrix ensemble of real symmetric matrices with Gaussian-distributed entries, central to the study of eigenvalue statistics and universality in random matrix theory.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Gaussian Orthogonal Ensemble | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Gaussian random matrix ensemble
ⓘ
probability distribution on matrices ⓘ random matrix ensemble ⓘ |
| abbreviation | GOE ⓘ |
| application |
disordered systems
ⓘ
mesoscopic physics ⓘ multivariate statistics ⓘ nuclear physics ⓘ number theory ⓘ quantum chaos ⓘ statistical mechanics ⓘ |
| belongsTo | Wigner matrix ensembles NERFINISHED ⓘ |
| betaEnsembleParameter | 1 ⓘ |
| contrastWith |
Gaussian symplectic ensemble with beta equals 4
NERFINISHED
ⓘ
Gaussian unitary ensemble with beta equals 2 NERFINISHED ⓘ |
| diagonalEntryVariance | twice off-diagonal variance in standard normalization ⓘ |
| DysonIndex | 1 ⓘ |
| edgeEigenvalueDistribution | Tracy–Widom distribution for beta equals 1 ⓘ |
| eigenvalueDensity | Wigner semicircle law NERFINISHED ⓘ |
| eigenvalueStatistics | Wigner–Dyson statistics NERFINISHED ⓘ |
| entryDistribution | Gaussian distribution ⓘ |
| field |
mathematical physics
ⓘ
probability theory ⓘ random matrix theory ⓘ |
| hasSymmetryGroup | orthogonal group O(n) NERFINISHED ⓘ |
| introducedBy | Eugene Wigner NERFINISHED ⓘ |
| invarianceProperty | invariant under orthogonal conjugation ⓘ |
| jointEigenvalueDensity | given by Vandermonde determinant to the first power times Gaussian weight ⓘ |
| levelRepulsionExponent | 1 ⓘ |
| matrixEntryMean | 0 ⓘ |
| matrixEntryVariance | depends on normalization convention ⓘ |
| matrixType | real symmetric matrices ⓘ |
| probabilityDensityForm | proportional to exp of minus n over 4 times trace of H squared in standard normalization ⓘ |
| relatedConcept |
Dyson Brownian motion
NERFINISHED
ⓘ
Gaussian symplectic ensemble NERFINISHED ⓘ Gaussian unitary ensemble NERFINISHED ⓘ orthogonal group NERFINISHED ⓘ |
| scalingLimit |
Airy kernel at the soft edge
NERFINISHED
ⓘ
sine kernel in the bulk ⓘ |
| spectralMeasureLimit | semicircle distribution ⓘ |
| symmetryClass | orthogonal symmetry ⓘ |
| timeReversalSymmetry | present ⓘ |
| typicalMatrixSize | n by n real symmetric matrix ⓘ |
| typicalNormalization | entries scaled by 1 over square root of matrix size ⓘ |
| universalityRole | prototype for universality of eigenvalue statistics ⓘ |
| usedFor | modeling spectra of time-reversal invariant quantum systems ⓘ |
| usedIn | universality proofs for Wigner matrices ⓘ |
Referenced by (2)
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this entity surface form:
Gaussian Orthogonal Ensemble