Jacobi ensemble
E865110
The Jacobi ensemble is a family of random matrix models whose eigenvalue distributions are supported on a finite interval and are closely connected to classical orthogonal polynomials and beta-type probability measures.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
probability distribution family
ⓘ
random matrix ensemble ⓘ |
| arisesFrom |
eigenvalues of MANOVA-type random matrices
ⓘ
eigenvalues of ratios of Wishart matrices ⓘ |
| dependsOn |
Dyson index β>0
ⓘ
matrix dimension n ⓘ shape parameters α>-1, γ>-1 ⓘ |
| generalizes | classical Jacobi weight ⓘ |
| hasAlternativeName | Jacobi β-ensemble NERFINISHED ⓘ |
| hasApplication |
quantum transport in mesoscopic systems
ⓘ
random eigenvalue problems with boundary constraints ⓘ |
| hasConnectionTo |
Coulomb gas models on a finite interval
ⓘ
Selberg integral NERFINISHED ⓘ integrable probability ⓘ |
| hasEdgeScalingLimit |
hard-edge universality
ⓘ
soft-edge universality (in appropriate regimes) ⓘ |
| hasEigenvalueDistribution | joint density supported on a compact interval ⓘ |
| hasJointEigenvalueDensityProportionalTo | ∏_{i<j} |λ_i-λ_j|^β ∏_{i} λ_i^{α} (1-λ_i)^{γ} ⓘ |
| hasLimitingBehavior | global eigenvalue density converges to a deterministic limit as matrix size grows ⓘ |
| hasLimitingKernel |
Airy-type kernels at the soft edge
ⓘ
Bessel-type kernels at the hard edge ⓘ sine kernel in the bulk (for β=1,2,4 under standard scaling) ⓘ |
| hasOrthogonalPolynomialSystem | Jacobi polynomials with respect to w(x)=x^{α}(1-x)^{γ} ⓘ |
| hasParameter |
α (shape parameter)
ⓘ
β (Dyson index) ⓘ γ (shape parameter) ⓘ |
| hasSpecialCase |
orthogonal Jacobi ensemble (β=1)
NERFINISHED
ⓘ
symplectic Jacobi ensemble (β=4) NERFINISHED ⓘ unitary Jacobi ensemble (β=2) ⓘ |
| hasSupport |
[0,1] (after affine rescaling)
ⓘ
finite interval ⓘ |
| hasSymmetryClass | orthogonal/unitary/symplectic depending on β ⓘ |
| isAssociatedWith |
beta distributions
ⓘ
beta-type measures ⓘ orthogonal polynomials ⓘ random matrix theory ⓘ |
| isCharacterizedBy | Vandermonde determinant to the power β in joint eigenvalue density ⓘ |
| isConnectedTo | Jacobi polynomials NERFINISHED ⓘ |
| isDefinedOn | space of eigenvalues λ_1,…,λ_n in (0,1) ⓘ |
| isPartOf | β-ensembles in random matrix theory ⓘ |
| isRelatedTo |
Gaussian ensemble
NERFINISHED
ⓘ
Laguerre ensemble NERFINISHED ⓘ |
| isUsedIn |
multivariate statistics
ⓘ
signal processing ⓘ wireless communications ⓘ |
| isUsedToModel |
canonical correlations in multivariate analysis
ⓘ
eigenvalue statistics of truncated unitary matrices ⓘ |
| usesWeightFunction | w(x)=x^{α}(1-x)^{γ} on [0,1] ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.