Nearest-neighbour spacing distributions of the β-Hermite ensemble of random matrices

Caër, G. Le; Male, Camille; Delannay, Renaud

doi:10.1016/j.physa.2007.04.057

Cited by 29 publications

(29 citation statements)

References 67 publications

(92 reference statements)

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“…However, it is clear that gamma distributions do not precisely model the analytic systems discussed here, and do not give correct asymptotic behaviour at the origin, as is evident from the results of Caër et al [11] who obtained excellent approximations for GOE, GUE and GSE distributions using the generalized gamma distribution (1.2) . The differences may be seen in Figure 3 which shows the unit mean distributions for gamma (dashed) and generalized gamma [11] (solid) fits to the true variances for the Poisson, GOE, GUE and GSE ensembles. Table 2: Effect of sample size: Statistical data for spacings in ten blocks of increasing size 200, 000m, m = 1, 2, .…”

Section: Remarkmentioning

confidence: 79%

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A Note on Quantum Chaology and Gamma Approximations to Eigenvalue Spacings for Infinite Random Matrices

Dodson

2009

Topics on Chaotic Systems

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Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We use known bounds on the distribution function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices and show that gamma distributions, which have an important uniqueness property, can yield an approximation to the GOE distribution. That has the advantage that then both chaotic and non chaotic cases fit in the information geometric framework of the manifold of gamma distributions, which has been the subject of recent work on neighbourhoods of randomness for general stochastic systems. Additionally, gamma distributions give approximations, to eigenvalue spacings for the Gaussian unitary ensemble (GUE) of infinite random hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite random hermitian matrices with real quaternionic elements, except near the origin. Gamma distributions do not precisely model the various analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes.

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Section: Remarkmentioning

confidence: 79%

“…Then the best fits of (1.2) had the parameter values [11] and were accurate to within ∼ 0.1% of the true distributions from Forrester [15]. Observe that the exponential distribution is recovered by the choice g(s; 0, 1) = e −s .…”

Section: Gsementioning

confidence: 84%

A Note on Quantum Chaology and Gamma Approximations to Eigenvalue Spacings for Infinite Random Matrices

Dodson

2009

Topics on Chaotic Systems

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“…For β ∈ R + \ {1, 2, 4} no such precise numerical schemes for the evaluation of p β are available. Instead, we use the generalised Wigner surmise (see [22]), which is only an approximation to the limiting distribution. One Looking at (40) one notes that the numerics will not detect the replacement of s −∞ dµ β by an approximation s −∞ dμ β as long as their deviation is small compared to E N .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Remark 4.9. We note that for the results presented in section 5.1 it is not necessary to keep track of the k-dependence of the error in (22). However, this estimate is needed in the proof of Theorem 6.2.…”

Section: Universality Of the K-point Correlation Functions For Invarimentioning

confidence: 99%

Spacings: An Example for Universality in Random Matrix Theory

Kriecherbauer¹,

Schubert²

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the empirical spacing distribution and its Kolmogorov distance from the universal limit. We describe new results, some analytical, some numerical, that are contained in [27]. A large part of the paper is devoted to explain basic definitions and facts of Random Matrix Theory, culminating in a sketch of the proof of a weak version of convergence for the empirical spacing distribution σN (see (23)).

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“…The raw level spacing has already been computed in the context of a 2 × 2 β-Hermite ensemble in (40).…”

Section: Spectral Propertiesmentioning

confidence: 99%

On invariant 2×2 β -ensembles of random matrices

Vivo

Majumdar

2008

Physica A: Statistical Mechanics and its Applications

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We introduce and solve exactly a family of invariant 2×2 random matrices, depending on one parameter η, and we show that rotational invariance and real Dyson index β are not incompatible properties. The probability density for the entries contains a weight function and a multiple trace-trace interaction term, which corresponds to the representation of the Vandermonde-squared coupling on the basis of power sums. As a result, the effective Dyson index β eff of the ensemble can take any real value in an interval. Two weight functions (Gaussian and non-Gaussian) are explored in detail and the connections with β-ensembles of Dumitriu-Edelman and the so-called Poisson-Wigner crossover for the level spacing are respectively highlighted. A curious spectral twinning between ensembles of different symmetry classes is unveiled: as a consequence, the identification between symmetry group (orthogonal, unitary or symplectic) and the exponent of the Vandermonde (β = 1, 2, 4) is shown to be potentially deceptive. The proposed technical tool more generically allows for designing actual matrix models which i) are rotationally invariant; ii) have a real Dyson index β eff ; iii) have a pre-assigned confining potential or alternatively level-spacing profile. The analytical results have been checked through numerical simulations with Preprint submitted to Elsevier 8 March 2018an excellent agreement. Eventually, we discuss possible generalizations and further directions of research.

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Nearest-neighbour spacing distributions of the β-Hermite ensemble of random matrices

Cited by 29 publications

References 67 publications

A Note on Quantum Chaology and Gamma Approximations to Eigenvalue Spacings for Infinite Random Matrices

A Note on Quantum Chaology and Gamma Approximations to Eigenvalue Spacings for Infinite Random Matrices

Spacings: An Example for Universality in Random Matrix Theory

On invariant 2×2 β -ensembles of random matrices

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