Generalized Ensemble of Random Matrices

Moshe, Moshe; Neuberger, Herbert; Shapiro, Boris

doi:10.1103/physrevlett.73.1497

Cited by 116 publications

(195 citation statements)

References 31 publications

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“…The random matrix model of Moshe and Neuberger and Shapiro [40] is defined by the partition function…”

Section: Definition Of the Modelmentioning

confidence: 99%

“…As we will see below, in order to make contact with the Thouless energy in the partially quenched effective partition function we have to scale h with an additional factor 1/ √ N. The unitary invariance of this partition function follows from the invariance of the Haar measure. In comparison to [40], an additional factor ∼ Σ 2 /h has been included in the probability distribution of the matrix elements. As we will see below, this will guarantee that in the thermodynamic limit the spectral density is h-independent to leading order in h. We will also find that the partition function is normalized such that the Σ represents the chiral condensate by means of the Banks-Casher relation Σ = πρ(0)/N (with ρ(0) the spectral density around λ = 0).…”

Section: Definition Of the Modelmentioning

confidence: 99%

“…This raises the question whether the Dirac eigenvalues might be described by critical statistics. To address this issue we generalize a random matrix model for critical statistics [40,41] to include the chiral symmetry of the QCD partition function (section 2). In section 3 we map our model on a partition function of noninteracting fermions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Chiral random matrix model for critical statistics

García-García

Verbaarschot

2000

Nuclear Physics B

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We propose a random matrix model that interpolates between the chiral random matrix ensembles and the chiral Poisson ensemble. By mapping this model on a noninteracting Fermi-gas we show that for energy differences less than a critical energy E c the spectral correlations are given by chiral Random Matrix Theory whereas for energy differences larger than E c the number variance shows a linear dependence on the energy difference with a slope that depends on the parameters of the model. If the parameters are scaled such that the slope remains fixed in the thermodynamic limit, this model provides a description of QCD Dirac spectra in the universality class of critical statistics. In this way a good description of QCD Dirac spectra for gauge field configurations given by a liquid of instantons is obtained.

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“…The random matrix model of Moshe and Neuberger and Shapiro [40] is defined by the partition function…”

Section: Definition Of the Modelmentioning

confidence: 99%

Section: Definition Of the Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Chiral random matrix model for critical statistics

García-García

Verbaarschot

2000

Nuclear Physics B

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“…On a more formal note we can hope the problems related to critical statistics [50,52] can also be attacked in this way, and indeed specialists in the field have expressed interest to use the power map on chiral ensembles and to regularize Dirac operators [53].…”

Section: Discussionmentioning

confidence: 99%

Emerging spectra of singular correlation matrices under small power-map deformations

2013

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Correlation matrices are a standard tool in the analysis of the time evolution of complex systems in general and financial markets in particular. Yet most analysis assume stationarity of the underlying time series. This tends to be an assumption of varying and often dubious validity. The validity of the assumption improves as shorter time series are used. If many time series are used this implies an analysis of highly singular correlation matrices. We attack this problem by using the so called power map which was introduced to reduce noise. Its non-linearity breaks the degeneracy of the zero eigenvalues and we analyze the sensitivity of the so emerging spectra to correlations. This sensitivity will be demonstrated for uncorrelated and correlated Wishart ensembles.

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“…Typical features include: scale invariant spectrum [20], level repulsion, and sub-Poisson number variance [21]. Different generalized random matrix model (gRMM) have been successfully employed to describe critical statistics [22,23].…”

Section: Introductionmentioning

confidence: 99%

Long range disorder and Anderson transition in systems with chiral symmetry

García-García

Takahashi

2004

Nuclear Physics B

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We study the spectral properties of a chiral random banded matrix (chRBM) with elements decaying as a power-law Hij ∼ |i−j| −α . This model is equivalent to a chiral 1D Anderson Hamiltonian with long range power-law hopping. In the weak disorder limit we obtain explicit nonperturbative analytical results for the density of states (DoS) and the two-level correlation function (TLCF) by mapping the chRBM onto a nonlinear σ model. We also put forward, by exploiting the relation between the chRBM at α = 1 and a generalized chiral random matrix model, an exact expression for the above correlation functions. We give compelling analytical and numerical evidence that for this value the chRBM reproduces all the features of an Anderson transition. Finally we discuss possible applications of our results to quantum chromodynamics (QCD).

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Generalized Ensemble of Random Matrices

Cited by 116 publications

References 31 publications

Chiral random matrix model for critical statistics

Chiral random matrix model for critical statistics

Emerging spectra of singular correlation matrices under small power-map deformations

Long range disorder and Anderson transition in systems with chiral symmetry

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