Matrix models for beta ensembles

By Ioana Dumitriu, Alan Edelman. 2002

723 citations recorded by the source corpus.

This paper constructs tridiagonal random matrix models for general (β>0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β=1,2,4. Furthermore, in the cases of the β-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.

Subfields: Applied Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d49ea8d2bddc9080a53f97e4ef18dff2cb6afae3

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In Applied Mathematics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles (2009, background): “Several results of this genre have been proved, though only for systems which lead to β ∈ {1, 2, 4}.”
Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices (2012, methodology): “For each (n,p) combination, we show in the first row empirical cumulative probabilities for λ1, rescaled by (4), and the second row, with parentheses, rescaled by (3), both computed from R= 40000 repeated draws from Wp(n, I) using the method in [7].”
Matrix models for circular ensembles (2004, background): “(5.12) We do not have a direct derivation of this fact.”
Edge Universality of Beta Ensembles (2013, background): “Furthermore, for general β in the Gaussian case, i.e., when V is quadratic, Dumitriu and Edelman [18] proved that the measure (1.1) describes the eigenvalue distribution of a special tridiagonal matrix.”
Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles (2009, background): “Edelman [28] proved that the expected value of the empirical distribution converges.”
Random matrix theory in statistics: A review (2014, background): “…of models for which explicit Tracy–Widom-type limit laws for the extreme eigenvalues have been found is the class of β-ensembles (Dumitriu, 2003; Dumitriu and Edelman, 2002) which correspond to random matrices that can be expressed as WWn, where W is a bi-diagonal matrix with independent…”
The importance of the Selberg integral (2007, methodology): “The methods of [39] and [92], which at a technical level proceed via a change of variables from a general tridiagonal matrix and unitary Hessenberg matrix to their eigenvalue/eigenvector decomposition, yield too the evaluations of the Mehta and Selberg integrals respectively.”
Asymptotic Expansion of β Matrix Models in the One-cut Regime (2011, background): “l symmetric (resp. hermitian, and quaternionic self-dual) matrices [Meh04]. For general >0, when V is quadratic (Hermite weight), or log + linear (Laguerre weight), Dumitriu and Edelman have found [DE02] a measure on a set of tridiagonal matrices which induces the measure dV;R N; on eigenvalues. ”
Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations (2004, neutral): “We restrict ourselves to those three cases in this paper, and refer the reader to [3] for recent results on the general β case.”
Random matrix theory (2019, background): “(See the seminal paper by Dumitriu and Edelman [8], and also papers by Edelman and Sutton [arxiv:0607038], Ramirez, Valkó, and Virág [arxiv:0607331v4], and Valkó and Virág [arxiv:0712.”
The local relaxation flow approach to universality of the local statistics for random matrices (2009, background): “, the Laguerre ensemble corresponds to the Gaussian sample covariance matrices which are also called Wishart matrices), see [11,23] for more details.”
Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles (2006, background): “Since the construction ofthe bbi isrecursive,itispossible (for xed n)to considerthe whole sequence ofvariables fdetX n;r;r= 1;:::;ng atthe sam etim e. Itcorrespondsto the decom position ofthe determ inantofa sym m etricpositive m atrix A as detA = rY j= 1 detA [j] detA [j 1] ; where A [j] is the j…”
From Random Matrices to Stochastic Operators (2006, neutral): “ar X iv :m at h- ph /0 60 70 38 v2 7 S ep 2”