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.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Jacobi ensemble canonical | 1 |
| Jacobi β-ensemble | 1 |
| Jacobi β-ensembles | 1 |
| beta-Jacobi ensembles | 1 |
| orthogonal Jacobi ensemble (β=1) | 1 |
| symplectic Jacobi ensemble (β=4) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462109 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Jacobi ensemble Context triple: [Selberg integral, relatedTo, Jacobi ensemble]
-
A.
Gaussian orthogonal ensemble
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.
-
B.
Gaussian symplectic 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.
-
C.
Gaussian unitary ensemble
The Gaussian unitary ensemble is a fundamental random matrix ensemble of complex Hermitian matrices with statistically independent, Gaussian-distributed entries, central to quantum chaos and random matrix theory.
-
D.
Wishart distribution
The Wishart distribution is a fundamental probability distribution over positive-definite matrices that generalizes the chi-squared distribution to multiple dimensions and underpins many multivariate statistical methods.
-
E.
Wigner surmise
The Wigner surmise is an approximate formula in random matrix theory that describes the statistical distribution of spacings between neighboring energy levels in complex quantum systems.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Jacobi ensemble Target entity description: 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.
-
A.
Gaussian orthogonal ensemble
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.
-
B.
Gaussian symplectic 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.
-
C.
Gaussian unitary ensemble
The Gaussian unitary ensemble is a fundamental random matrix ensemble of complex Hermitian matrices with statistically independent, Gaussian-distributed entries, central to quantum chaos and random matrix theory.
-
D.
Wishart distribution
The Wishart distribution is a fundamental probability distribution over positive-definite matrices that generalizes the chi-squared distribution to multiple dimensions and underpins many multivariate statistical methods.
-
E.
Wigner surmise
The Wigner surmise is an approximate formula in random matrix theory that describes the statistical distribution of spacings between neighboring energy levels in complex quantum systems.
- F. None of above. chosen
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
ⓘ
linked to:
Jacobi ensemble
|
| hasApplication |
quantum transport in mesoscopic systems
ⓘ
random eigenvalue problems with boundary constraints ⓘ |
| hasConnectionTo |
Coulomb gas models on a finite interval
ⓘ
Selberg integral ⓘ 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)
ⓘ
linked to:
Jacobi ensemble
symplectic Jacobi ensemble (β=4) ⓘ
linked to:
Jacobi ensemble
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 ⓘ |
| isDefinedOn | space of eigenvalues λ_1,…,λ_n in (0,1) ⓘ |
| isPartOf | β-ensembles in random matrix theory ⓘ |
| isRelatedTo |
Gaussian ensemble
ⓘ
linked to:
Gaussian ensembles
Laguerre ensemble ⓘ
linked to:
Laguerre β-ensembles
|
| 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] ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jacobi ensemble Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Jacobi ensemble
linked to: Jacobi ensemble
linked to: Jacobi ensemble
linked to: Jacobi ensemble
linked to: Jacobi ensemble