Finite‐size corrections at the hard edge for the Laguerre β ensemble

Forrester, Peter J.; Trinh, Allan K.

doi:10.1111/sapm.12279

Cited by 21 publications

(18 citation statements)

References 41 publications

(166 reference statements)

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“…12 & 13], which are well suited to the analysis of the rate of convergence to the hard edge limit. This circumstance similarly holds true for the Laguerre case of (A.1), for which an analysis of the rate of convergence has recently been carried out in [23].…”

Section: Appendix Amentioning

confidence: 56%

“…Recalling (1.6), we see from (3.10) that in general for products of Laguerre unitary ensembles, the pointwise rate of convergence to the hard edge limiting k-point correlation is O(1/N ). On the other hand, as noted in the text around (1.11), earlier works [8,11,23,27,43] have demonstrated that for the Laguerre unitary ensemble itself (the case M = 1), with the hard edge scaling variables as used in (3.10), and with the Laguerre parameter a = 0, the convergence rate is actually O(1/N 2 ). Moreover, these same references found that the O(1/N 2 ) rate holds for general Laguerre parameter a > −1 if each N on the LHS of (3.10) is replaced by N + a/2.…”

Section: 1mentioning

confidence: 91%

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Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices

Forrester¹,

Li²

2020

Preprint

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The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large N asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib-Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes as a special case the Jacobi unitary ensemble.In each case the hard edge scaled kernel permits an expansion in powers of 1/N , with the leading term given in a structured form involving the hard edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembles have the special feature that their hard edge scaled kernel -the Bessel kernel -is symmetric and this leads to there being a choice of hard edge scaling variables for which the rate of convergence of the correlation functions is O(1/N 2 ).

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Section: Appendix Amentioning

confidence: 56%

Section: 1mentioning

confidence: 91%

Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices

Forrester¹,

Li²

2020

Preprint

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“…Remark 2.4. The analytic structure of an exponential times a polynomial has been present, and played an important role, in a number of recent studies of distributions in random matrix theory [36,22,27].…”

Section: Generalising (115)mentioning

confidence: 99%

Joint moments of a characteristic polynomial and its derivative for the circular $β$-ensemble

Forrester

2020

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The problem of calculating the scaled limit of the joint moments of the characteristic polynomial, and the derivative of the characteristic polynomial, for matrices from the unitary group with Haar measure first arose in studies relating to the Riemann zeta function in the thesis of Hughes. Subsequently, Winn showed that these joint moments can equivalently be written as the moments for the distribution of the trace in the Cauchy unitary ensemble, and furthermore relate to certain hypergeometric functions based on Schur polynomials, which enabled explicit computations. We give a β-generalisation of these results, where now the role of the Schur polynomials is played by the Jack polynomials. This leads to an explicit evaluation of the scaled moments for all β > 0, subject to the constraint that a particular parameter therein is equal to a non-negative integer. Consideration is also given to the calculation of the moments of the singular statistic ∑ N j=1 1/x j for the Jacobi β-ensemble.

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“…A recent application of the differential-difference system to the computation of some structured formulas for the distribution of the smallest and largest eigenvalues of the β-Jacobi ensemble, applicable in certain parameter ranges, has been given in [14]. In the case of the β-Laguerre ensemble analogue of (2.17), the works [24,13,17] exhibit analogous applications. For present purposes, a particular change of variables and limit of (2.17) and (2.18) is required.…”

Section: The Differential-difference System Let C Nmentioning

confidence: 99%

Differential recurrences for the distribution of the trace of the $β$-Jacobi ensemble

Forrester,

Kumar

2020

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Examples of the β-Jacobi ensemble specify the joint distribution of the transmission eigenvalues in scattering problems. In this context, there has been interest in the distribution of the trace, as the trace corresponds to the conductance. Earlier, in the case β = 1, the trace statistic was isolated in studies of covariance matrices in multivariate statistics, where it is referred to as Pillai's V statistic. In this context, Davis showed that for β = 1 the trace statistic, and its Fourier-Laplace transform, can be characterised by (N + 1) × (N + 1) matrix differential equations. For the Fourier-Laplace transform, this leads to a vector recurrence for the moments. However, for the distribution itself the characterisation provided was incomplete, as the connection problem of determining the linear combination of Frobenius type solutions that correspond to the statistic was not solved. We solve this connection problem for Jacobi parameter b and Dyson index β non-negative integers. For the other Jacobi parameter a also a non-negative integer, the power series portion of each Frobenius solution terminates to a polynomial, and the matrix differential equation gives a recurrence for their computation.

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Finite‐size corrections at the hard edge for the Laguerre β ensemble

Cited by 21 publications

References 41 publications

Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices

Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices

Joint moments of a characteristic polynomial and its derivative for the circular $β$-ensemble

Differential recurrences for the distribution of the trace of the $β$-Jacobi ensemble

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