## Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models (2006)

Venue: | J. Math. Phys |

Citations: | 10 - 2 self |

### BibTeX

@ARTICLE{Dumitriu06globalspectrum,

author = {Ioana Dumitriu and Alan Edelman},

title = {Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models},

journal = {J. Math. Phys},

year = {2006},

volume = {47}

}

### OpenURL

### Abstract

ensembles via matrix models

### Citations

840 | Symmetric functions and Hall polynomials - Macdonald - 1979 |

637 |
Random Matrices
- Mehta
- 1991
(Show Context)
Citation Context ...ine, or the unit circle in the complex plane; other possibilities have been considered, too, and generalizations are easily conceived. A good reference for these formulae can be found in Mehta’s book =-=[29]-=-. Some of the most studied eigenvalue ensembles have Hermite, Laguerre, and Jacobi weight functions on the real line, or uniform weight on the unit circle. In this paper we will be examining the ensem... |

515 |
Aspects of Multivariate Statistical Theory
- Muirhead
- 1982
(Show Context)
Citation Context ...ing the ensembles with Hermite and Laguerre weights on the real line (respectively, half-line); see Table 1. For more references on Gaussian ensembles, see [29]; for Wishart and MANOVA ensembles, see =-=[30]-=-; for Hermite, Laguerre, and Jacobi ensembles, see [16]. For three particular values of β, namely 1, 2, and 4, these ensembles have been studied since the birth of the field, as the Gaussian real, com... |

237 | Shape fluctuations and random matrices - Johansson |

178 |
Distributions of eigenvalues for some sets of random matrices
- Marcenko, Pastur
- 1967
(Show Context)
Citation Context ...ase of Laguerre ensembles, 2a/(nβ) → γ, the particles have an emerging (global) level density which is obeys a simple law (Wigner’s semicircle law [45] for the Hermite ensembles, Marčenko-Pastur laws =-=[28]-=- for the Laguerre ensembles). The roots of the Hermite, respectively Laguerre, polynomial have this same asymptotical density – the fluctuations do not change the asymptotics, as they are on a smaller... |

172 |
On the distributions of the roots of certain symmetric matrices
- Wigner
- 1958
(Show Context)
Citation Context ...ghtly larger classes of random matrices. 1 Introduction 1.1 The Semicircle Law, deviations and fluctuations, numerically The most celebrated theorem of random matrix theory, the Wigner semicircle law =-=[44, 45]-=-, may be illustrated as in Figure 1 by histogramming the eigenvalues of a single random symmetric matrix using the simple MATLAB code (normalization omitted) A = randn(n); S = (A + A ′ )/sqrt(8 ∗ n); ... |

165 |
Characteristic vectors of bordered matrices with infinite dimensions
- Wigner
- 1955
(Show Context)
Citation Context ...ghtly larger classes of random matrices. 1 Introduction 1.1 The Semicircle Law, deviations and fluctuations, numerically The most celebrated theorem of random matrix theory, the Wigner semicircle law =-=[44, 45]-=-, may be illustrated as in Figure 1 by histogramming the eigenvalues of a single random symmetric matrix using the simple MATLAB code (normalization omitted) A = randn(n); S = (A + A ′ )/sqrt(8 ∗ n); ... |

151 | On orthogonal and symplectic matrix ensembles
- Tracy, Widom
- 1996
(Show Context)
Citation Context ...erties we enumerate the level spacings for the Gaussian ensembles (see [29], [39]) and the extremal (largest, corresponding to “soft edge”, smallest, to “hard edge”) eigenvalue asymptotics (see [41], =-=[40]-=-, [22], [23]). All these results relate to real, complex, or quaternion matrices; in the category of results relating to general β, we have to mention recent work by Desrosiers and Forrester [8], wher... |

150 |
Distributions of matrix variates and latent roots derived from normal samples
- James
- 1964
(Show Context)
Citation Context ...[1]). Similarly, the Wishart real and complex (Laguerre with β = 1, 2 and some restrictions on the Laguerre parameter) matrices emerged from the world of statistical multivariate analysis ([46], [5], =-=[20]-=-, [24]). The parameter β (making the connection to the Boltzmann factor of statistical physics) is seen by some communities (e.g. statistical mechanics) as an inverse temperature, or repulsion strengt... |

136 |
Some combinatorial properties of Jack symmetric functions
- Stanley
- 1989
(Show Context)
Citation Context ...complex, and quaternion entries in the matrix models. However, some communities (like algebraic combinatorics) consider a different parameter, α = 2/β, which tends to simplify certain formulas ([27], =-=[36]-=-). In this paper we will use both notations, for convenience, and make sure that the reader is informed when changes take place. The reason for the attractivity and success that the study of Gaussian ... |

111 |
On fluctuations of eigenvalues of random Hermitian matrices
- Johansson
- 1998
(Show Context)
Citation Context ...ext order behavior in the law, for large n, we can subtract away the semicircle and multiply by n. The next order average behavior is what we call the deviation and it was first computed by Johansson =-=[21]-=- to be DEV IAT ION = 1 4 δ−1(x) + 1 4 δ1(x) − 1 � 1 − x2 ; (1) 2π here ∼ stands for “has the distribution of”. This expression for the deviation is the β = 1 (corresponding to real matrices) instance ... |

99 |
H.: Fredholm determinants, differential equations and matrix models
- Tracy, Widom
- 1994
(Show Context)
Citation Context ...to refer to a property that occurs near an individual or a constant number of eigenvalues. Among the more famous local properties we enumerate the level spacings for the Gaussian ensembles (see [29], =-=[39]-=-) and the extremal (largest, corresponding to “soft edge”, smallest, to “hard edge”) eigenvalue asymptotics (see [41], [40], [22], [23]). All these results relate to real, complex, or quaternion matri... |

94 |
Some limit theorems for the eigenvalues of a sample covariance matrix
- Jonsson
- 1982
(Show Context)
Citation Context ...Similarly, the Wishart real and complex (Laguerre with β = 1, 2 and some restrictions on the Laguerre parameter) matrices emerged from the world of statistical multivariate analysis ([46], [5], [20], =-=[24]-=-). The parameter β (making the connection to the Boltzmann factor of statistical physics) is seen by some communities (e.g. statistical mechanics) as an inverse temperature, or repulsion strength, of ... |

89 |
The Generalised Product Moment Distribution in Samples from a Normal Multivariate Population
- Wishart
- 1928
(Show Context)
Citation Context ...45], [14], [1]). Similarly, the Wishart real and complex (Laguerre with β = 1, 2 and some restrictions on the Laguerre parameter) matrices emerged from the world of statistical multivariate analysis (=-=[46]-=-, [5], [20], [24]). The parameter β (making the connection to the Boltzmann factor of statistical physics) is seen by some communities (e.g. statistical mechanics) as an inverse temperature, or repuls... |

86 | Matrix models for beta ensembles
- Dumitriu, Edelman
(Show Context)
Citation Context ...atrix models Ioana Dumitriu and Alan Edelman October 10, 2005 Abstract We study the global spectrum fluctuations for β-Hermite and β-Laguerre ensembles via the tridiagonal matrix models introduced in =-=[11]-=-, and prove that the fluctuations describe a Gaussian process on monomials. We extend our results to slightly larger classes of random matrices. 1 Introduction 1.1 The Semicircle Law, deviations and f... |

69 | The Calogero-Sutherland model and generalized classical polynomials
- BAKER, FORRESTER
- 1997
(Show Context)
Citation Context ... [10] as Theorem 8.5.3 (the proof is virtually the same as in [10] and we will not repeat it here). The second one is a rewrite of a particular case (za = 0 and formula (4.14a)) of formula (4.36b) in =-=[3]-=-. Lemma 2.6. Let κ ′ denote the conjugate partition to κ (obtained by transposing the rows and columns in the Young tableau). Then the following is true: � � � Jα κ (x1, . . ., xn) . E H α J α κ (In) ... |

60 | Large deviations for Wigner’s law and Voiculescu’s noncommutative entropy, Prob - Arous, Guionnet - 1997 |

49 |
Some noncentral distribution problems in multivariate analysis
- Constantine
- 1963
(Show Context)
Citation Context ...14], [1]). Similarly, the Wishart real and complex (Laguerre with β = 1, 2 and some restrictions on the Laguerre parameter) matrices emerged from the world of statistical multivariate analysis ([46], =-=[5]-=-, [20], [24]). The parameter β (making the connection to the Boltzmann factor of statistical physics) is seen by some communities (e.g. statistical mechanics) as an inverse temperature, or repulsion s... |

48 | On the distribution of the largest principal component
- Johnstone
- 2000
(Show Context)
Citation Context ...umerate the level spacings for the Gaussian ensembles (see [29], [39]) and the extremal (largest, corresponding to “soft edge”, smallest, to “hard edge”) eigenvalue asymptotics (see [41], [40], [22], =-=[23]-=-). All these results relate to real, complex, or quaternion matrices; in the category of results relating to general β, we have to mention recent work by Desrosiers and Forrester [8], where they analy... |

48 | Matrix models for circular ensembles
- Killip, Nenciu
(Show Context)
Citation Context ...(statistical properties of the eigenvalues). One of the developments in the study of arbitrary β-Hermite, -Laguerre, and -Jacobi ensembles is the introduction of general real matrix models (see [11], =-=[25]-=-, [38]). For every β, there are simple real tridiagonal matrices which model the corresponding eigenvalue distributions given by Table 1. For the β-Hermite and -Laguerre ensembles, we present these fo... |

46 | Oscillation matrices and kernels and small vibrations of mechanical systems - Gantmakher, Krein - 2002 |

44 |
The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics
- Dyson
- 1962
(Show Context)
Citation Context ... namely 1, 2, and 4, these ensembles have been studied since the birth of the field, as the Gaussian real, complex, and quaternion ensembles (Hermite with β = 1, 2, 4) of nuclear physics ([44], [45], =-=[14]-=-, [1]). Similarly, the Wishart real and complex (Laguerre with β = 1, 2 and some restrictions on the Laguerre parameter) matrices emerged from the world of statistical multivariate analysis ([46], [5]... |

39 |
Large deviations asymptotics for spherical integrals
- Guionnet, Zeitouni
- 2002
(Show Context)
Citation Context ...f interest is represented by asymptotical large deviations from the density (spectral measure); we mention the results of [4], [2], [18] (which also covers global fluctuations for Wishart matrices) , =-=[19]-=-, [7] (which covers moderate deviations). For the linear statistic �n i=1 f(λi), the Wigner and Marčenko-Pastur laws for the β-Hermite and β-Laguerre ensembles state that for any “well-behaved” functi... |

37 | CLT for linear spectral statistics of large dimensional sample covariance matrices. Annals of Probability 32
- Bai, Silverstein
- 2004
(Show Context)
Citation Context ...β-Laguerre ensembles (see Table 2). General β results for this kind of statistic can be found in [6], [21]; this or similar linear statistics have been considered also in [15] (for unitary matrices), =-=[32]-=- (for Wishart matrices), and, heuristically, in [31]. Another path of interest is represented by asymptotical large deviations from the density (spectral measure); we mention the results of [4], [2], ... |

34 |
A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices, Functional Anal
- Sinai
- 1998
(Show Context)
Citation Context ...ns, perturbation theory, and combinatorial pathcounting. Some of the techniques we used in studying the traces of powers of random matrices have been inspired by the work of Soshnikov and Sinai [33], =-=[34]-=-, and by the study of traces of unitary random matrices by Diaconis and Shahshahani [9]. 1.3 Statements of results Among the asymptotical eigenstatistics one may distinguish two classes of properties:... |

30 | The distribution of largest eigenvalue in the Gaussian ensembles, in Calogero-Moser-Sutherland Models
- Tracy, Widom
- 2000
(Show Context)
Citation Context ...l properties we enumerate the level spacings for the Gaussian ensembles (see [29], [39]) and the extremal (largest, corresponding to “soft edge”, smallest, to “hard edge”) eigenvalue asymptotics (see =-=[41]-=-, [40], [22], [23]). All these results relate to real, complex, or quaternion matrices; in the category of results relating to general β, we have to mention recent work by Desrosiers and Forrester [8]... |

27 |
Large deviation upper bounds and central limit theorems for band matrices
- Guionnet
- 1999
(Show Context)
Citation Context ... (for Wishart matrices), and, heuristically, in [31]. Another path of interest is represented by asymptotical large deviations from the density (spectral measure); we mention the results of [4], [2], =-=[18]-=- (which also covers global fluctuations for Wishart matrices) , [19], [7] (which covers moderate deviations). For the linear statistic �n i=1 f(λi), the Wigner and Marčenko-Pastur laws for the β-Hermi... |

25 |
On the statistical mechanics approach in the random matrix theory: integrated density of states
- Monvel, Pastur, et al.
- 1995
(Show Context)
Citation Context ...the scaled eigenvalues λi. The scaling is λ → √ 2nβλ for the β-Hermite ensembles and λ → nβ/γλ for the β-Laguerre ensembles (see Table 2). General β results for this kind of statistic can be found in =-=[6]-=-, [21]; this or similar linear statistics have been considered also in [15] (for unitary matrices), [32] (for Wishart matrices), and, heuristically, in [31]. Another path of interest is represented by... |

23 |
Fluctuations de la loi empirique de grandes matrices aléatoires
- Cabanal-Duvillard
(Show Context)
Citation Context ...se results are new in the general β context; the global spectrum fluctuations of complex Wishart matrices (β-Laguerre with β = 2) were studied by Speicher et al. [26], and before by Cabanal-Duvillard =-=[4]-=-. Johansson’s more extensive study [21] covers our results for the β-Hermite ensembles. To prove our theorems, we use a very diverse set of methods and techniques from Jack Polynomial theory, special ... |

22 | Eigenvalue statistics for the Beta-ensembles
- Dumitriu
- 2003
(Show Context)
Citation Context ...guerre ensembles). The roots of the Hermite, respectively Laguerre, polynomial have this same asymptotical density – the fluctuations do not change the asymptotics, as they are on a smaller scale. In =-=[10]-=-, we have proved convergence almost surely (as n → ∞) of the asymptotical eigenvalue distribution of the β-Hermite ensemble to the semicircle distribution S with density 2 √ π 1 − x2 , and of the asym... |

17 |
On Wigner’s semicircle law for the eigenvalues of random matrices
- Arnold
- 1971
(Show Context)
Citation Context ...y 1, 2, and 4, these ensembles have been studied since the birth of the field, as the Gaussian real, complex, and quaternion ensembles (Hermite with β = 1, 2, 4) of nuclear physics ([44], [45], [14], =-=[1]-=-). Similarly, the Wishart real and complex (Laguerre with β = 1, 2 and some restrictions on the Laguerre parameter) matrices emerged from the world of statistical multivariate analysis ([46], [5], [20... |

13 |
The Stochastic Operator Approach to Random Matrix Theory, PhD thesis at MIT, available at “http://faculty.rmc.edu/bsutton
- Sutton
(Show Context)
Citation Context ...stical properties of the eigenvalues). One of the developments in the study of arbitrary β-Hermite, -Laguerre, and -Jacobi ensembles is the introduction of general real matrix models (see [11], [25], =-=[38]-=-). For every β, there are simple real tridiagonal matrices which model the corresponding eigenvalue distributions given by Table 1. For the β-Hermite and -Laguerre ensembles, we present these forms in... |

11 |
Central limit theorems for traces of large symmetric matrices with independent matrix elements
- Sinai, Soshnikov
- 1998
(Show Context)
Citation Context ...unctions, perturbation theory, and combinatorial pathcounting. Some of the techniques we used in studying the traces of powers of random matrices have been inspired by the work of Soshnikov and Sinai =-=[33]-=-, [34], and by the study of traces of unitary random matrices by Diaconis and Shahshahani [9]. 1.3 Statements of results Among the asymptotical eigenstatistics one may distinguish two classes of prope... |

10 | Eigenvalues of Hermite and Laguerre Ensembles: Large Beta Asymptotics. Annales de l’Institut Henri Poincare
- Dumitriu, Edelman
- 2005
(Show Context)
Citation Context ...verse temperature). As β → ∞ the particle positions 5 ⎞ ⎟ ⎠ ⎞sbehave like multivariate normals with variance O(1/β) and means located at the roots of a Hermite (respectively Laguerre) polynomial (see =-=[12]-=-). For a fixed β, as n → ∞ and, in the case of Laguerre ensembles, 2a/(nβ) → γ, the particles have an emerging (global) level density which is obeys a simple law (Wigner’s semicircle law [45] for the ... |

10 | MOPS: Multivariate Orthogonal Polynomials (symbolically). 2004. Preprint found at lanl.arxiv.org/abs/mathph/0409066
- Dumitriu, Edelman, et al.
(Show Context)
Citation Context ...k polynomial theory allows us to take a complicated random matrix problem and reduce it to an exercise on the properties of univariate Hermite and Laguerre polynomials. 2. With the Maple Library MOPs =-=[13]-=-, we can compute symbolically the exact values of momentk(n) for small values of k, as a function of n and β. In other words, while this paper concerns itself with the constant and O(1/n) behavior, it... |

9 | Hermite and Laguerre β-ensembles: asymptotic corrections to the eigenvalue density
- Desrosiers, Forrester
(Show Context)
Citation Context ...41], [40], [22], [23]). All these results relate to real, complex, or quaternion matrices; in the category of results relating to general β, we have to mention recent work by Desrosiers and Forrester =-=[8]-=-, where they analyze the asymptotical corrections to the eigenvalue density, and, for β ∈ 2N, obtain the expected O(n 2/3 ) order in the fluctuation of the largest eigenvalue in the case of both β-Her... |

7 | Orthogonal Polynomials and Fluctuations of Random Matrices
- Kusalik, Mingo, et al.
- 2005
(Show Context)
Citation Context ...aguerre ensembles. For the latter, these results are new in the general β context; the global spectrum fluctuations of complex Wishart matrices (β-Laguerre with β = 2) were studied by Speicher et al. =-=[26]-=-, and before by Cabanal-Duvillard [4]. Johansson’s more extensive study [21] covers our results for the β-Hermite ensembles. To prove our theorems, we use a very diverse set of methods and techniques ... |

7 |
Hermite Polynomial.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HermitePolynomial.html
- Weisstein
(Show Context)
Citation Context ...h1/ √ 2n, . . ., hn/ √ 2n, where h1, . . ., hn are the roots of the nth Hermite polynomial Hn(x) (this can be easily deduced from the threeterm recurrence for the Hermite polynomials, see for example =-=[42]-=-). For a more detailed description of the properties of this matrix, see [12]. It follows that the generating function we need to compute is We use the well-known identity ˜m(n, x) = 1 n n� i=1 1 x − ... |

6 |
Random-matrix description of the distribution of mesoscopic conductance
- Politzer
- 1989
(Show Context)
Citation Context ...s for this kind of statistic can be found in [6], [21]; this or similar linear statistics have been considered also in [15] (for unitary matrices), [32] (for Wishart matrices), and, heuristically, in =-=[31]-=-. Another path of interest is represented by asymptotical large deviations from the density (spectral measure); we mention the results of [4], [2], [18] (which also covers global fluctuations for Wish... |

5 | Moderate Deviations for the Spectral Measure of certain Random Matrices
- Dembo, Guionnet, et al.
- 2003
(Show Context)
Citation Context ...rest is represented by asymptotical large deviations from the density (spectral measure); we mention the results of [4], [2], [18] (which also covers global fluctuations for Wishart matrices) , [19], =-=[7]-=- (which covers moderate deviations). For the linear statistic �n i=1 f(λi), the Wigner and Marčenko-Pastur laws for the β-Hermite and β-Laguerre ensembles state that for any “well-behaved” function f,... |

4 |
Combinatorial Aspects of Free Probability Theory
- Speicher
- 2005
(Show Context)
Citation Context ...he mean and scales by the variance to obtain a limiting Gaussian, are sometimes called central limit theorems (see for example [32] and [33]). Free Probability uses this term as well, see for example =-=[35]-=-, in a different context, namely, to express the fact that averaging over random matrices creates an eigenvalue distribution that approaches the semi-circular law. Both uses of the term “central limit... |

3 |
Global fluctuation formulas and universal correlations for random matrices and log-gas systems at infinite density. Nuclear Phys
- Forrester
- 1995
(Show Context)
Citation Context ...embles and λ → nβ/γλ for the β-Laguerre ensembles (see Table 2). General β results for this kind of statistic can be found in [6], [21]; this or similar linear statistics have been considered also in =-=[15]-=- (for unitary matrices), [32] (for Wishart matrices), and, heuristically, in [31]. Another path of interest is represented by asymptotical large deviations from the density (spectral measure); we ment... |

1 |
CLToflinearspectralstatistics oflargedimensional sample covariance matrices. preprint, 2003. Accepted for publication in Annals. of Probab
- Bai
(Show Context)
Citation Context ...β-Laguerre ensembles (see Table 2). General β results for this kind of statistic can be found in [6], [21]; this or similar linear statistics have been considered also in [15] (for unitary matrices), =-=[32]-=- (for Wishart matrices), and, heuristically, in [31]. Another path of interest is represented by asymptotical large deviations from the density (spectral measure); we mention the results of [4], [2], ... |