## Limiting laws of linear eigenvalue statistics for unitary invariant matrix models (2006)

Venue: | J. Math. Phys |

Citations: | 6 - 3 self |

### BibTeX

@ARTICLE{Pastur06limitinglaws,

author = {L. Pastur},

title = {Limiting laws of linear eigenvalue statistics for unitary invariant matrix models},

journal = {J. Math. Phys},

year = {2006},

pages = {103303}

}

### OpenURL

### Abstract

We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n ×n Hermitian matrices as n → ∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of q ≥ 2 intervals, then in the global regime the variance of statistics is a quasiperiodic function of n as n → ∞ generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not in general 1/2 × variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.

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Citation Context ...ntial V in (2.20) by V/g, g > 0, introducing explicitly the amplitude of the potential. Then the quantities of asymptotic formulas (2.38) – (2.41) will depend on g, and it follows from the results of =-=[17, 26]-=- that these quantities will be continuous functions of g in a certain neighborhood of g = 1, provided that the support (1.7) for g = 1 consists The components of the vector β = {βl} q−1 l=1 (g). Takin... |

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Citation Context ... (in the sense (1.10)) Laplace transform is not quadratic in ϕ, hence the limiting law is not Gaussian (see formulas (3.1), (3.13), and (3.19) – (3.20)). This has to be compared with results of paper =-=[14]-=-, according to which the limits of variance and the probability law are the same for all sequences nj → ∞ (i.e., exist), and the limiting probability law is Gaussian. 3The random matrix theory deals ... |

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Citation Context ...mble Nn → N. (1.5) The corresponding scale (asymptotic regime) is called the global (or macroscopic). The convergence is either in probability or even with probability 1. We refer the reader to works =-=[19, 9, 30]-=-, where this fact is proved and discussed for two most widely studied classes of random matrix ensembles, the Wigner Ensembles (independent or weakly dependent entries) and the Matrix Models (invarian... |

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Citation Context ...nique minimizer of (2.50). Then, according to [12] (see also [11, 29]), the non-increasing function ν(λ) = ν([λ, ∞)) (cf (2.38)) coincides with the function, defined in (2.46). Moreover, according to =-=[31]-=- the measure ν is the Integrated Density of States (IDS) measure of J(x) (see [32] for a definition of the IDS measure in a general setting of ergodic operators), the support (1.7) is spectrum of J(x)... |

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1 |
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1 |
Random Matrices (New
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(Show Context)
Citation Context ... statistics (1.2) and for the Laplace transform of their probability law via special orthogonal polynomials. The technique dates back to works by Dyson, Gaudin, Mehta, and Wigner of the 60s (see e.g. =-=[27]-=-). Namely, we have for the joint probability density of eigenvalues of ensemble (2.1): where and pn(λ1, ..., λn) = ψ (n) l ( det { ψ (n) j−1 (λk) } ) n 2/ n!, (2.20) j,k=1 = e−nV/2 P (n) l , (2.21) {P... |