## A Simple Approach To Global Regime Of The Random Matrix Theory (1999)

Venue: | In Mathematical results in statistical mechanics |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Pastur99asimple,

author = {Leonid Pastur},

title = {A Simple Approach To Global Regime Of The Random Matrix Theory},

booktitle = {In Mathematical results in statistical mechanics},

year = {1999},

pages = {429--454},

publisher = {World Sci. Publishing}

}

### OpenURL

### Abstract

. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1. Introduction Random matrix theory is actively developing. Among numerous topics of the theory and its various applications those related to the asymptotic eigenvalue distribution of random matrices of large order are of considerable interest. An important role in this branch of the theory plays the eigenvalue counting measure defined for any Hermitian or real symmetr...

### Citations

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Random Matrices
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Citation Context ...here \Delta is a Borel set of the real axis R and f (n) i g n i=1 are eigenvalues of M n . One distinguishes several large-n asymptotic regimes for the probability properties of eigenvalues (see e.g. =-=[1]-=-). In this paper we deal with the global regime, defined by the requirement that the expectation E(N n (\Delta)) has well defined (i.e. not zero and not infinite) weak limit N (\Delta) = lim n!1 E(N n... |

626 |
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Citation Context ...ndependent Gaussian entries is well known since the 30's in the multivariate analysis as the Wishart distribution and describes the sample covariance matrix of m random Gaussian n-dimensional vectors =-=[13]-=-. Theorem 2.2. Let the Laguerre ensemble of random matrices be defined as above. Then its eigenvalue counting measure converges in probability 1 to the nonrandom measure of the form N L (\Delta) = 1 4... |

316 |
Free random variables
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Citation Context ...f [14]. (iii). The simple differentiation formula (2.45) as well as its version (4.5) below allows one to give a direct proof of the asymptotic freeness of unitary and diagonal matrices as n ! 1 (see =-=[2]-=- for definitions and results and [16, 26, 17] for some related recent results). Existing proofs are based on the representation of the Haar distributed unitary random matrices U as the phase in the po... |

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Citation Context ...6) where the symbol v:p: R denotes the singular Cauchy integral. Regarding this relation as a singular integral equation for ae() and using standard facts of the theory of singular integral equations =-=[21]-=- we find that the bounded solution of the equation has the form ae() = 1 2 p R() Z b a V 0 () \Gamma V 0 ()s\Gammasd p R() ; R() = (b \Gamma )( \Gamma a); (3.17) provided that Z b a V 0 ()d p R() = 0:... |

127 | Theory of Linear Operators in Hilbert Space - Akhiezer, Glazman - 1993 |

110 |
Statistical Mechanics
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Citation Context ...mz q jsy; q = 1; ::: for some y ? 0. Then relations (2.12) and (2.19) imply the bounds jg n (z)js1 jImzj 1 y ; (2.22) jm p js1 y p ; jr p js2w 2 p n 2 y p+1 : Following statistical mechanics (see e.g.=-=[5]-=-) we can treat system of identities (2.21) as a linear equation in the Banach space B of complex valued sequences m of functions m = fm p (z 1 ; :::; z p )g 1 p=1 equipped with the norm jjmjj = sup p1... |

99 | Quantum field theory techniques in graphical enumeration, Hdv - Bessis, Itzykson, et al. - 1980 |

55 | Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles
- Pastur, Shcherbina
(Show Context)
Citation Context ...ortunately, we do not know the proof of the last estimate based on the differentiation formula (3.9) and similar to that in the second proof of Proposition 2.1. Thus we refer the reader to works [19],=-=[20]-=-, where the bound (3.6) is proven for all Imzj ? 0 by using a combination of the orthogonal polynomials and variational techniques. A simple proof of a weaker version of (3.6) with n instead n 2 in th... |

44 |
ed): Free Probability Theory
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(Show Context)
Citation Context ...ation formula (2.45) as well as its version (4.5) below allows one to give a direct proof of the asymptotic freeness of unitary and diagonal matrices as n ! 1 (see [2] for definitions and results and =-=[16, 26, 17]-=- for some related recent results). Existing proofs are based on the representation of the Haar distributed unitary random matrices U as the phase in the polar decomposition of the Gaussian distributed... |

43 |
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Citation Context ...e counting measures were used in [14] in the qualitative study of the support of of these measures in the case (5.16) below where R(f) = \Gammac Z toe(dt) 1 + tf : (4.11) Relation (4.10) was noted in =-=[28]-=- for the case when NA and NB are both the semicircle laws (2.5), when R(f) = w 2 f . The general form of this relation was proposed by D.Voiculescu in the context of the operator-algebras theory and i... |

26 |
On the statistical mechanics approach to the random matrix theory: the integrated density of states
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(Show Context)
Citation Context .... Unfortunately, we do not know the proof of the last estimate based on the differentiation formula (3.9) and similar to that in the second proof of Proposition 2.1. Thus we refer the reader to works =-=[19]-=-,[20], where the bound (3.6) is proven for all Imzj ? 0 by using a combination of the orthogonal polynomials and variational techniques. A simple proof of a weaker version of (3.6) with n instead n 2 ... |

21 | 2D gravity and random matrices
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Citation Context ...'s. As it was in the case of the Gaussian Ensembles we discuss here the technically simplest Hermitian matrices. This subclass of ensembles (3.1) is motivated by Quantum Field Theory (see e.g. review =-=[18]-=-). Following Quantum Field Theory we will 1 Although one can always use the probability space that is the product over all n of the probability spaces consisting of the groups U(n) with the normalized... |

20 | Limiting eigenvalue distribution of random matrices with correlated entries. Markov Process
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- 1996
(Show Context)
Citation Context ...2 \Gamma z) (2.28) where the radical is defined by the condition that it behaves as z as z ! 1. By using the inversion formula (2.9) we obtain (2.5). Theorem 2.1 is proved. The method was proposed in =-=[6]-=- and subsequently used in [7, 8, 9, 10] to study a variety of problem of random matrix theory and its applications. A certain disadvantage of the method is that it is rather tedious. An important simp... |

15 |
A random matrix model from two-dimensional YangMills theory
- Xu
- 1997
(Show Context)
Citation Context ...ation formula (2.45) as well as its version (4.5) below allows one to give a direct proof of the asymptotic freeness of unitary and diagonal matrices as n ! 1 (see [2] for definitions and results and =-=[16, 26, 17]-=- for some related recent results). Existing proofs are based on the representation of the Haar distributed unitary random matrices U as the phase in the polar decomposition of the Gaussian distributed... |

14 |
Pastur, â€śLimits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness
- Khorunzhy, A
- 1993
(Show Context)
Citation Context ...radical is defined by the condition that it behaves as z as z ! 1. By using the inversion formula (2.9) we obtain (2.5). Theorem 2.1 is proved. The method was proposed in [6] and subsequently used in =-=[7, 8, 9, 10]-=- to study a variety of problem of random matrix theory and its applications. A certain disadvantage of the method is that it is rather tedious. An important simplification of the method was proposed b... |

12 |
Eigenvalue distribution in some ensembles of random matrices
- Marchenko, Pastur
- 1967
(Show Context)
Citation Context ...oofs in this paper are based on the study the Stieltjes transforms of measures instead of measures themselves. In the random matrix theory the Stieltjes transform was used for the first time in paper =-=[14]-=- and since then is proved to be a rather efficient tool of the study of the global regime. Recall that the Stieltjes transform f(z) of a non-negative measure m(d); m(R) = 1; is the function of the com... |

10 |
Spectral Theory of Random Matrices
- Girko
- 1988
(Show Context)
Citation Context ...niscent the well known Lindeberg condition of the validity of the central limit theorem. This fact is known since the seventies, see [28] for the sufficiency of somewhat stronger version of (5.8) and =-=[29, 30]-=- for the necessity and sufficiency of (5.8). However these results were obtained by rather complicated method. In [3] we give a simple proof based on the approach of this paper. (ii) 1/n expansion [31... |

9 |
Eigenvalue distribution of large random matrices with correlated entries
- Khorunzhy
- 1996
(Show Context)
Citation Context ...riety of problem of random matrix theory and its applications. A certain disadvantage of the method is that it is rather tedious. An important simplification of the method was proposed by A.Khorunzhy =-=[11]-=-. We describe now this simpler version by giving another proof of the previous proposition. Rewrite the first two equations of the system (2.21) for z 1 = z; z 2 =sz in the form E(g n (z)) = \Gamma 1 ... |

8 |
On the universality of the level spacing distribution for some ensembles of random matrices
- Pastur
- 1992
(Show Context)
Citation Context ...e I ff = Z 1 0 t ff dt p 1 \Gamma t 2 These formulas are also valid for non-integer p, i.e. for potentials of the form V () = jj ff =ff provided that ffs2. For this case the formulas were obtained in =-=[22] by a-=-nother method. They can also be obtained by a version of the method presented in this Section. In this version we use the identity G(z)V 0 (M) = G(z)V 0 () +G(z)(V 0 (M) \Gamma V 0 ()); z =s+ i" ... |

7 |
The large-n limit in statistical mechanics and the spectral theory of disordered systems
- Khorunzhy, Khoruzhenko, et al.
- 1992
(Show Context)
Citation Context ...radical is defined by the condition that it behaves as z as z ! 1. By using the inversion formula (2.9) we obtain (2.5). Theorem 2.1 is proved. The method was proposed in [6] and subsequently used in =-=[7, 8, 9, 10]-=- to study a variety of problem of random matrix theory and its applications. A certain disadvantage of the method is that it is rather tedious. An important simplification of the method was proposed b... |

7 |
Free convolution and the random sum of matrices
- Speicher
- 1993
(Show Context)
Citation Context ...Gamma 1 where f belongs to the class (2.8) and \Delta A and \Delta B are analytic for non-real z and such that sup y1 yj\Delta A (iy)j ! 1; sup y1 yj\Delta B (iy)j ! 1 (4.4) The theorem was proved in =-=[25]-=- for the case of uniformly bounded in n matrices A n and B n by computing asymptotic form of moments of the sum via the moments of summands. This requires rather involved combinatorial analysis and im... |

7 |
On the law of multiplication of random matrices
- Vasilchuk
(Show Context)
Citation Context ...ula (5.16) below) plays the role of the Poisson distribution. (iv). Similar technique can be applied to multiplicative families of positive defined Hermitian and or unitary matrices and gives results =-=[27]-=- that generalize and simplify those of [14] and also gives a more direct proof of certain results obtained for these ensembles in the context of free probability [2]. 5. Matrices With Independent and ... |

6 |
Eigenvalue distribution for band random matrices in the limit of their infinite rank
- Molchanov, Pastur, et al.
- 1992
(Show Context)
Citation Context ...radical is defined by the condition that it behaves as z as z ! 1. By using the inversion formula (2.9) we obtain (2.5). Theorem 2.1 is proved. The method was proposed in [6] and subsequently used in =-=[7, 8, 9, 10]-=- to study a variety of problem of random matrix theory and its applications. A certain disadvantage of the method is that it is rather tedious. An important simplification of the method was proposed b... |

6 | On the law of addition of random matrices
- Pastur, Vasilchuk
(Show Context)
Citation Context ... X(XX ) \Gamma1=2 these proofs are not simple to implement in all details. The approach based on the formula (4.5), i.e. on the shift invariance of the Haar measure, seems more direct and simple (see =-=[23]-=- and Remark (iv) of Section 4). 3. Invariant Ensembles In this Section we discuss the random matrix ensembles defined by the probability law P (dM) = Z \Gamma1 n exp(\Gamman Tr V (M))dM (3.1) where M ... |

5 |
and L.A.Pastur: The Interband Light Absorption Coefficient in the weak disorder regime: An asymptotically exactly solvable model
- Khoruzenko, Kirsch
- 1994
(Show Context)
Citation Context |

3 |
of addition in random matrix theory
- Zee
- 1996
(Show Context)
Citation Context ...m of two independent matrices distributed each according to the law (3.1) with possibly different polynomials V 1;2 . In this case the condition (4.2) follows from (3.15). This case was considered in =-=[24]-=- by using formal perturbation theory around the Gaussian ensemble. Thus, we see that Theorem 4.1 describes in A SIMPLE APPROACH TO GLOBAL REGIME OF THE RANDOM MATRIX THEORY 17 a rather general setting... |

3 |
Rigorous mean-field model for coherent-potential approximation: Anderson model with free random variables
- Neu, Speicher
- 1995
(Show Context)
Citation Context ...ation formula (2.45) as well as its version (4.5) below allows one to give a direct proof of the asymptotic freeness of unitary and diagonal matrices as n ! 1 (see [2] for definitions and results and =-=[16, 26, 17]-=- for some related recent results). Existing proofs are based on the representation of the Haar distributed unitary random matrices U as the phase in the polar decomposition of the Gaussian distributed... |

1 |
Random transfer matrix theory and conductance fluctuations
- Pichard
- 1991
(Show Context)
Citation Context ... n are independent complex Gaussian random variables defined by E(A jk ) = E(A 2 jk ) = 0; E(jA jk j 2 ) = 2a 2 : Note that the matrix A is not Hermitian. The name of the ensemble is recent (see e.g. =-=[12]-=-) and is related to the fact that in the orthogonal polynomial approach [1] one uses in the case of this ensemble the Laguerre polynomials (recall that in the case of the A SIMPLE APPROACH TO GLOBAL R... |

1 |
Random matrices with independent entries: asymptotic properties of the Green function
- Khorunzhy, Khoruzhenko, et al.
- 1996
(Show Context)
Citation Context ...not necessary Gaussian and may be n-dependent. We call these 18 LEONID PASTUR ensembles the Wigner Ensembles. In this general case we have to use instead of differentiation formula (2.18) the formula =-=[31] E('()) = -=-s X a=1sa+1 a! E(' (a) ()) + " s (5.2) wheresa are semi-invariants (cumulants) of a real-valued random variable , ' : R ! C is a function of the class C s+1 and j" s jsC s sup x j' (s+1) (x)... |