Computable structural formulas for the distribution of the $β$-Jacobi edge eigenvalues

Forrester, Peter J.; Kumar, Santosh

doi:10.48550/arxiv.2006.02238

Cited by 2 publications

(2 citation statements)

References 31 publications

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“…An alternative method of derivation is also possible, as we will now show, which involves first identifying differential recurrences satisfied by each of the p n (x) and q n (x). (We remark that other examples of differential recurrences can be found in a number of recent studies in random matrix theory [15,16,19,20,35].) Proposition 2.1.…”

Section: 1mentioning

confidence: 81%

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: 1mentioning

confidence: 81%

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

Forrester¹,

Li²

2020

Preprint

Self Cite

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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.…”

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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Computable structural formulas for the distribution of the $β$-Jacobi edge eigenvalues

Cited by 2 publications

References 31 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

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

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