The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities

Soshnikov, Alexander

doi:10.1214/aop/1019160338

Cited by 138 publications

(164 citation statements)

References 32 publications

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“…This theorem can be viewed as an extension of results obtained in Jonsson (1982) where the entries of X n are Gaussian and T n = I and is consistent with central limit theorem results on linear statistics of eigenvalues of other classes of random matrices [see, e.g., Johansson (1998), Sinai and Soshnikov (1998), Soshnikov (2000) and Diaconis and Evans (2001)]. As will be seen, the techniques and arguments used to prove the theorem, which rely heavily on properties of the Stieltjes transform of F B n , have nothing in common with any of the tools used in these other papers.…”

supporting

confidence: 75%

CLT for Linear Spectral Statistics of Large-Dimensional Sample Covariance Matrices

Bai

Silverstein

2008

Advances in Statistics

142

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The limiting behavior, as n → ∞ with n/N approaching a positive constant, of functionals of the eigenvalues of B n , where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of B n , it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be 1/n by proving, after proper scaling, that they form a tight sequence. Moreover, if EX 2 11 = 0 and E|X 11 | 4 = 2, or if X 11 and T n are real and EX 4 11 = 3, they are shown to have Gaussian limits.

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supporting

confidence: 75%

CLT for Linear Spectral Statistics of Large-Dimensional Sample Covariance Matrices

Bai

Silverstein

2008

Advances in Statistics

142

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“…For mesoscopic linear statistics, to our knowledge, Theorem 1.1 only appeared for β = 2 in a paper of Soshnikov [46]. Soshnikov's method is very different from ours: it relies on the method of moments and it does not yield the convergence of the Laplace transform of a linear statistics as in Theorem 1.1.…”

Section: This Implies Thatmentioning

confidence: 86%

“…Let us mention that for β = 2, there is another coupling between the CUE and Sine 2 existed prior to [51,48] which is based on virtual isometries [7]. Moreover, it is possible to obtain Theorem 1.12 directly by using the determinantal structure of the Sine 2 process, see Kac [30] and Soshnikov [46]. Finally, it should be mentioned that there have recently been several developments in the study of the Sine β for general β > 0.…”

Section: Clt For the Sine β Point Processesmentioning

confidence: 99%

Mesoscopic central limit theorem for the circular $\beta $-ensembles and applications

Lambert¹

2021

Electron. J. Probab.

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We give a simple proof of a central limit theorem for linear statistics of the circular β-ensembles which is valid at almost microscopic scales for functions of class C 3 . Using a coupling introduced by Valkò and Viràg [48], we deduce a central limit theorem for the Sine β processes. We also discuss connections between our result and the theory of Gaussian Multiplicative Chaos. Based on the results of [37], we show that the exponential of the logarithm of the real (and imaginary) part of the characteristic polynomial of the circular β-ensembles, regularized at a small mesoscopic scale and renormalized, converges to GMC measures in the subcritical regime. This establishes that the leading order behavior for the extreme values of the logarithm of the characteristic polynomial is consistent with the predictions coming from log-correlated Gaussian field theory.

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“…Recalling that the conductance in chaotic cavities can be indeed written as a linear statistics of a random matrix (see below), the phenomenon of UCF is readily understood. The issue of fluctuations of generic linear statistics has however a longer history in the physics and mathematics literature [6][7][8][9][10][11][12][13][14][15], due to its relevance for a variety of applications beyond UCF, ranging from quantum transport in metallic conductors [16] and entanglement of trapped fermion chains [17] to the statistics of extrema of disordered landscapes [18] to mention just a few.…”

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confidence: 99%

Universal Covariance Formula for Linear Statistics on Random Matrices

Cunden

Vivo

2014

Phys. Rev. Lett.

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We derive an analytical formula for the covariance Cov(A, B) of two smooth linear statistics A = i a(λi) and B = i b(λi) to leading order for N → ∞, where {λi} are the N real eigenvalues of a general one-cut random-matrix model with Dyson index β. The formula, carrying the universal 1/β prefactor, depends on the random-matrix ensemble only through the edge points [λ−, λ+] of the limiting spectral density. For A = B, we recover in some special cases the classical variance formulas by Beenakker and Dyson-Mehta, clarifying the respective ranges of applicability. Some choices of a(x) and b(x) lead to a striking decorrelation of the corresponding linear statistics. We provide two applications -the joint statistics of conductance and shot noise in ideal chaotic cavities, and some new fluctuation relations for traces of powers of random matrices.Introduction -The discovery of the phenomenon of universal conductance fluctuations (UCF) in disordered metallic samples, pioneered by Altshuler [1] and Lee and Stone [2] has had a profound impact on our current understanding of the mechanisms of quantum transport at low temperatures and voltage. There are two aspects of this universality, i.) the variance of the conductance is of order (e 2 /h) 2 , independent of sample size or disorder strength, and ii.) this variance decreases by precisely a factor of two if time-reversal symmetry is broken by a magnetic field. Both features, observed in several experiments and numerical simulations (see [3] for a review), naturally emerge from a random-matrix theoretical formulation of the electronic transport problem [4,5]. The phenomenon of UCF is just, however, one of the very many incarnations of a more general and intriguing property of sums of strongly correlated random variables.

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The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities

Abstract: We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn.

Cited by 138 publications

References 32 publications

CLT for Linear Spectral Statistics of Large-Dimensional Sample Covariance Matrices

CLT for Linear Spectral Statistics of Large-Dimensional Sample Covariance Matrices

Mesoscopic central limit theorem for the circular $\beta $-ensembles and applications

Universal Covariance Formula for Linear Statistics on Random Matrices

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