## On the Law of Addition of Random Matrices (2000)

Venue: | COMMUNICATIONS IN MATHEMATICAL PHYSICS |

Citations: | 6 - 1 self |

### BibTeX

@MISC{Pastur00onthe,

author = {L. Pastur and V. Vasilchuk},

title = {On the Law of Addition of Random Matrices},

year = {2000}

}

### OpenURL

### Abstract

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.

### Citations

682 |
Random Matrices
- Mehta
- 1991
(Show Context)
Citation Context ...stribution P(dM) = Z −1 { n exp − n TrM2} dM, (2.28) 4w2 dM = n∏ j=1 dMjj ∏ 1≤j<k≤n dRe MjkdImMjk, where Zn is the normalization constant. The distribution defines the Gaussian Unitary Ensemble (GUE) =-=[18]-=-. This is why ensemble (2.26) is called the deformed GUE [7]. It is known [18] that Mn can be written in the form Mn = U ∗ n ΛnUn, (2.29) where Un are unitary matrices whose probability law is the Haa... |

346 |
Probability theory
- Loéve
- 1955
(Show Context)
Citation Context ...esults of [27] proved under the condition that supports of the NCM Nr,n, r = 1, 2 of An and Bn are uniformly bounded in n. Remark 2 By mimicking the proof of the Glivenko - Cantelli theorem (see e.g. =-=[16]-=-), one can prove that the random distribution functions Nn(λ) = Nn(] − ∞, λ[) corresponding to measures (2.2) converge uniformly with probability 1 to the distribution function N(λ) = N(] − ∞, λ[) cor... |

246 | Spectra of Random and Almost-Periodic Operators - Pastur, Figotin - 1992 |

196 |
On the distribution of the roots of certain symmetric matrices, Ann
- Wigner
- 1958
(Show Context)
Citation Context ...trix identities, the resolvent identity first of all. The basic idea is the same as in [17, 20]: to study not the moments of the counting measure, as it was proposed in the pioneering paper by Wigner =-=[35]-=-, but rather its Stieltjes (called also the Cauchy or the Borel) transform, playing the role of appropriate generating (or characteristic) function of the moments (the measure). However, the technical... |

187 | Distribution of Eigenvalues for Some Sets of Random Matrices - Marchenko, Pastur - 1967 |

138 |
Limit laws for random matrices and free products
- Voiculescu
- 1991
(Show Context)
Citation Context ... developed in several directions (see e.g. [9] - [11] and the recent work [22]). Similar problems arose recently in operator algebras studies, known now as the free (non-commutative) probability (see =-=[29, 32, 30]-=- for results and references). In particular, the notion of the Rtransform and the free convolution of measures were introduced by Voiculescu and allowed the limiting eigenvalue distributions of the su... |

124 |
Das asymptotische Verteilungsgesetze der Eigenschwingungen eines beliebig gestalteten elastischen
- Weyl
- 1915
(Show Context)
Citation Context ...n the context of general problem to describe the eigenvalues of the sum of two matrices in terms of eigenvalues of two terms of the sum. The latter problem dates back at least to the paper of H. Weyl =-=[34]-=-, was treated in a number of papers, including the recent paper [15], and related to interesting questions of combinatorics, geometry, algebra etc. (see e.g. [8] for recent results and references). Th... |

121 | Eigenvalues, invariant factors, highest weigths, and Schubert calculus - Fulton - 2000 |

102 |
Stable bundles, representation theory and Hermitian operators, Sel
- Klyachko
- 1998
(Show Context)
Citation Context ...sum of two matrices in terms of eigenvalues of two terms of the sum. The latter problem dates back at least to the paper of H. Weyl [34], was treated in a number of papers, including the recent paper =-=[15]-=-, and related to interesting questions of combinatorics, geometry, algebra etc. (see e.g. [8] for recent results and references). The problem is also of considerable interest for mathematical physics ... |

87 |
Theory of Random Determinants
- Girko
- 1990
(Show Context)
Citation Context ...ts of Hermitian matrices. 4The Stieltjes transform was first used in studies of the eigenvalue distribution of random matrices in paper [17] and proved to be an efficient tool in the field (see e.g. =-=[9, 10, 11, 12, 13, 14, 20, 21, 22, 25, 26]-=-). We list the properties of the Stieltjes transform that we will need below (see e.g.[1]). Proposition 2.1 Let m be a non-negative and normalized to unity measure and ∫ m(dλ) s(z) = , Im z = 0 (2.11... |

87 | Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices
- Silverstein
- 1995
(Show Context)
Citation Context ...ts of Hermitian matrices. 4The Stieltjes transform was first used in studies of the eigenvalue distribution of random matrices in paper [17] and proved to be an efficient tool in the field (see e.g. =-=[9, 10, 11, 12, 13, 14, 20, 21, 22, 25, 26]-=-). We list the properties of the Stieltjes transform that we will need below (see e.g.[1]). Proposition 2.1 Let m be a non-negative and normalized to unity measure and ∫ m(dλ) s(z) = , Im z = 0 (2.11... |

65 |
Free convolution of measures with unbounded support
- Bercovici, Voiculescu
- 1993
(Show Context)
Citation Context ...2.20) were proposed in [17], where the respective functional equations analogous to (2.24) were derived. A general class of the random matrix ensembles of these forms were studied in free probability =-=[29, 32, 2]-=-, where the notions of the S - transform and the free multplicative convolution of measures were proposed and used to give a general form of the limiting eigenvalue distributions of products (2.38) an... |

55 | Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles
- Pastur, Shcherbina
(Show Context)
Citation Context ....2) and (2.3) and derive directly the functional equations for their limits and the bound analogous to (2.9) for the rate of their convergence (rather well known in the random matrix theory, see e.g. =-=[24, 11]-=-) by using certain simple identities for expectations of matrix functions with respect to the Haar measure (Proposition 3.2 below) and elementary facts on resolvents of Hermitian matrices. 4The Stiel... |

50 |
A strengthened asymptotic freeness result for random matrices with applications to free entropy
- Voiculescu
- 1998
(Show Context)
Citation Context ... measure (NCM) Nn of the ensemble (2.1), defined for any Borel set ∆ ⊂ R by the formula Nn(λ) = #{λi ∈ ∆} , (2.2) n where λi, i = 1, ..., n are the eigenvalues of Hn. The problem was studied recently =-=[32, 27, 31]-=- in the context of free (noncommutative) probability. In particular, it follows from results of [27] that if the matrices An and Bn are non-random, their norms are uniformly bounded in n, i.e. their N... |

45 | Analysis of the Limiting Spectral Distribution of Large Dimensional Information-Plus-Noise Type Matrices
- Dozier, Silverstein
(Show Context)
Citation Context ...ts of Hermitian matrices. 4The Stieltjes transform was first used in studies of the eigenvalue distribution of random matrices in paper [17] and proved to be an efficient tool in the field (see e.g. =-=[9, 10, 11, 12, 13, 14, 20, 21, 22, 25, 26]-=-). We list the properties of the Stieltjes transform that we will need below (see e.g.[1]). Proposition 2.1 Let m be a non-negative and normalized to unity measure and ∫ m(dλ) s(z) = , Im z = 0 (2.11... |

44 | ed): Free Probability Theory - Voiculescu - 1997 |

43 | The spectrum of random matrices - Pastur - 1972 |

40 | Asymptotic properties of large random matrices with independent entries
- Khorunzhy, Khoruzhenko, et al.
- 1996
(Show Context)
Citation Context ...tion of a certain functional equation. Thus, a randomized version of the problem admits a rather constructive and explicit solution. These results were developed in several directions (see e.g. [9] - =-=[11]-=- and the recent work [22]). Similar problems arose recently in operator algebras studies, known now as the free (non-commutative) probability (see [29, 32, 30] for results and references). In particul... |

26 |
On the statistical mechanics approach to the random matrix theory: the integrated density of states
- Monvel, Pastur, et al.
- 1995
(Show Context)
Citation Context ...m is related to the sum of irreducible diagrams of the formal perturbation series. Existence of the limiting eigenvalue counting measure for the random matrix ensemble (2.37) was rigorously proved in =-=[6]-=- for a rather broad class of functions V (not necessary polynomials). It was also proved that the normalized counting measure (2.2) converges in probability to the limiting measure. The form (2.29) of... |

24 | Random-matrix physics: spectrum and strength fluctuations - Brody, Flores, et al. - 1981 |

22 |
On the free convolution with a semi-circular distribution
- Biane
- 1997
(Show Context)
Citation Context ...erties of the limiting eigenvalue counting measure described by Theorem 2.1, i.e. the measure, whose Stieltjes transform is a solution of (2.18) satisfying (2.12)–(2.14). We refer the reader to works =-=[32, 2, 4, 3]-=- and references therein for a rather complete collection of results on properties of the measure, resulting from the binary operation in the space of the probability measures, defined by a version of ... |

20 | Limiting eigenvalue distribution of random matrices with correlated entries. Markov Process
- Monvel, Khorunzhy, et al.
- 1996
(Show Context)
Citation Context |

15 |
A random matrix model from two-dimensional YangMills theory
- Xu
- 1997
(Show Context)
Citation Context ...decomposition at zero. On the other hand, the differentiation formula (3.11) allows one to prove directly similar statements. Here is an example of results of this type (related results are proved in =-=[36]-=-). 35Theorem 6.1 Let k be a positive integer, {Tr,n} k r=1 be a set of n × nmatrices, such that sup n r≤k; k,l,n∈N −1 Tr(T ∗ r,nTr,n) l < ∞, (6.4) and let Un be the unitary and Haar-distributed rando... |

14 |
Pastur, “Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness
- Khorunzhy, A
- 1993
(Show Context)
Citation Context |

9 |
Glazman I.M., Theory of Linear Operators in Hilbert Space
- Akhiezer
- 1963
(Show Context)
Citation Context ...n paper [17] and proved to be an efficient tool in the field (see e.g. [9, 10, 11, 12, 13, 14, 20, 21, 22, 25, 26]). We list the properties of the Stieltjes transform that we will need below (see e.g.=-=[1]-=-). Proposition 2.1 Let m be a non-negative and normalized to unity measure and ∫ m(dλ) s(z) = , Im z = 0 (2.11) λ − z be the Stieltjes transform of m (here and below integrals without limits denote t... |

9 |
Eigenvalue distribution of large random matrices with correlated entries
- Khorunzhy
- 1996
(Show Context)
Citation Context |

7 |
J.-B.: Quantum field theory technique in graphical enumeration
- Bessis, Itzykson, et al.
- 1980
(Show Context)
Citation Context ...he normalized counting measure (2.2) converges in probability to the limiting measure. The form (2.29) of matrices of ensemble (2.37) can be deduced from known results on the ensemble (2.37) (see e.g.=-=[5]-=-) in the same way as for the GUE (2.28), where V (λ) = λ 2 /4w 2 (see [18]). Condition (2.17) follows from results of [6, 22]. Thus we can apply Theorem 2.1 to obtain rigorously relation (2.36) in the... |

7 |
The large-n limit in statistical mechanics and the spectral theory of disordered systems
- Khorunzhy, Khoruzhenko, et al.
- 1992
(Show Context)
Citation Context |

7 |
Free convolution and the random sum of matrices
- Speicher
- 1993
(Show Context)
Citation Context ... measure (NCM) Nn of the ensemble (2.1), defined for any Borel set ∆ ⊂ R by the formula Nn(λ) = #{λi ∈ ∆} , (2.2) n where λi, i = 1, ..., n are the eigenvalues of Hn. The problem was studied recently =-=[32, 27, 31]-=- in the context of free (noncommutative) probability. In particular, it follows from results of [27] that if the matrices An and Bn are non-random, their norms are uniformly bounded in n, i.e. their N... |

7 | On the law of multiplication of random matrices - Vasilchuk |

6 | R.: Rigorous mean field model for CPA: Anderson model with free random variables
- Neu, Speicher
- 1995
(Show Context)
Citation Context ...he Stieltjes transform of this limiting measure and fr(z) = ∫ ∞ −∞ N(dλ) , Imz > 0, (2.4) λ − z Nr(dλ) , r = 1, 2, (2.5) λ − z are the Stieltjes transforms of Nr, r = 1, 2 of (2.3), then according to =-=[19]-=- f(z) satisfies the functional equation f(z) = f1(z + R2(f(z))), (2.6) 3where R2(f) is defined by the relation z = − 1 f2(z) − R2(f2(z))) (2.7) and is known as R-transform of the measure N2 of (2.3) ... |

4 | On a simple approach to global regime of random matrix theory
- Pastur
- 1999
(Show Context)
Citation Context ...nal equation. Thus, a randomized version of the problem admits a rather constructive and explicit solution. These results were developed in several directions (see e.g. [9] - [11] and the recent work =-=[22]-=-). Similar problems arose recently in operator algebras studies, known now as the free (non-commutative) probability (see [29, 32, 30] for results and references). In particular, the notion of the Rtr... |

4 |
Free Probability Theory (Fields Institute
- Voiculescu
- 1997
(Show Context)
Citation Context ... developed in several directions (see e.g. [9] - [11] and the recent work [22]). Similar problems arose recently in operator algebras studies, known now as the free (non-commutative) probability (see =-=[29, 32, 30]-=- for results and references). In particular, the notion of the Rtransform and the free convolution of measures were introduced by Voiculescu and allowed the limiting eigenvalue distributions of the su... |

3 | Disordered system with N-orbitals per site-Lagrange formulation, hyperbolic symmetry, and Goldstone modes Z - Schaefer, Wegner - 1980 |

3 |
of addition in random matrix theory
- Zee
- 1996
(Show Context)
Citation Context ...2.36) for the case then matrices H1 and H2 distributed both according to the laws P (n) 1,2 (dH) = Z(n) 1,2 exp{−nV1,2(H)}dH. (2.37) where V1,2 : R → R+ are polynomials of an even degree was given in =-=[37]-=-. The derivation is based on the perturbation theory with respect to the non-quadratic part of V1,2 and the R-transform is related to the sum of irreducible diagrams of the formal perturbation series.... |

1 |
D.: Regularity questions for free convolution of measures
- Bercovici, Voiculescu
- 1998
(Show Context)
Citation Context ...erties of the limiting eigenvalue counting measure described by Theorem 2.1, i.e. the measure, whose Stieltjes transform is a solution of (2.18) satisfying (2.12)–(2.14). We refer the reader to works =-=[32, 2, 4, 3]-=- and references therein for a rather complete collection of results on properties of the measure, resulting from the binary operation in the space of the probability measures, defined by a version of ... |

1 | Random Matrices (Sluchainye matricy). Kiev: Vyshcha Shkola - Girko - 1975 |

1 |
A.: Free Probability Theory. A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups
- Voiculescu, Dykema, et al.
- 1992
(Show Context)
Citation Context ... developed in several directions (see e.g. [9] - [11] and the recent work [22]). Similar problems arose recently in operator algebras studies, known now as the free (non-commutative) probability (see =-=[29, 32, 30]-=- for results and references). In particular, the notion of the Rtransform and the free convolution of measures were introduced by Voiculescu and allowed the limiting eigenvalue distributions of the su... |

1 | Das asymptotischeVerteilungsgesetz der Eigenwerte lineare partieller differential Gleichungen - Weyl - 1912 |

1 |
Eigenvalues of sum of Hermitian matrices. In: Séminaire Bourbaki. Volume 1997/98. Exposes 835–849
- Fulton
- 1998
(Show Context)
Citation Context ...back at least to the paper of H. Weyl [34], was treated in a number of papers, including the recent paper [15], and related to interesting questions of combinatorics, geometry, algebra etc. (see e.g. =-=[8]-=- for recent results and references). The problem is also of considerable interest for mathematical physics because of its evident links with spectral theory and quantum mechanics (perturbation theory ... |

1 |
V.L.Random Matrices (Sluchainye matricy). Vyshcha Shkola
- Girko
- 1975
(Show Context)
Citation Context ...e solution of a certain functional equation. Thus, a randomized version of the problem admits a rather constructive and explicit solution. These results were developed in several directions (see e.g. =-=[9]-=- - [11] and the recent work [22]). Similar problems arose recently in operator algebras studies, known now as the free (non-commutative) probability (see [29, 32, 30] for results and references). In p... |

1 |
Pastur L.A.:, Distribution of eigenvalues for some sets of random matrices
- Marchenko
- 1967
(Show Context)
Citation Context ...ymptotic answer, studying a randomized version of the problem in which at least one of the two terms is random and both behave rather regularly as n → ∞. Particular results of this type were given in =-=[17, 20]-=- where it was proved that under certain conditions the divided by n eigenvalue counting measure of the sum converges in probability to the nonrandom limit that can be found as a unique solution of a c... |

1 | On the law of multilication of random matrices (submitted to - Vasilchuk |

1 |
Disordered systems with n-orbitals per site: n
- Wegner
- 1979
(Show Context)
Citation Context ...e sum of an arbitrary matrix and certain random matrices (see (2.20) and (2.26)), in particular, Gaussian random matrices (2.28). In this case, however, there exists another model, proposed by Wegner =-=[33]-=- that combines properties of random matrices, having all entries roughly of the same order, and of random operators, whose entries decay sufficiently fast in the distance from the principal diagonal (... |