Comment on “Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices”

Forrester, Peter J.; Trinh, Allan K.

doi:10.1103/physreve.99.036101

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…The normalisations appearing in (A. 19), (A.20) becomes M β (N + ã, , ¯2/β) as a result of the Morris integral (2.21).…”

Section: 4mentioning

confidence: 99%

“…With the parameters ã, b defined in (2.25), the numerator in (A. 19) has the β-dimensional integral representation given by…”

Section: 4mentioning

confidence: 99%

“…are well known in the context of the loop equation analysis of the Gaussian ensembles [39]. Moreover (1.3) can be extended to the Wigner class, with ρ W,1 (λ) now dependent on the fourth moment, but no higher moments; see the introduction to [19] for further discussion and references. The point to note then is that the rate of convergence in this setting holds true generally, while the density function at this order weakly depends on the details of the distribution of the entries.…”

Section: Introductionmentioning

confidence: 97%

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Asymptotic correlations with corrections for the circular Jacobi $β$-ensemble

Forrester¹,

Li²,

Trinh³

2020

Preprint

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Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their β generalisations at the hard and soft edge. It has been found that the functional form of this correction is given by a derivative operation applied to the leading term. In the present work we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble with Dyson indices β = 1, 2 and 4, and also to the spectral density in the corresponding β-ensemble with β even. The former requires an analysis involving the Routh-Romanovski polynomials, while the latter is based on multidimensional integral formulas for generalised hypergeometric series based on Jack polynomials. In all cases this correction term is found to be related to the leading term by a derivative operation.

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“…The normalisations appearing in (A. 19), (A.20) becomes M β (N + ã, , ¯2/β) as a result of the Morris integral (2.21).…”

Section: 4mentioning

confidence: 99%

“…With the parameters ã, b defined in (2.25), the numerator in (A. 19) has the β-dimensional integral representation given by…”

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

Asymptotic correlations with corrections for the circular Jacobi $β$-ensemble

Forrester¹,

Li²,

Trinh³

2020

Preprint

Self Cite

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“…where ρ W,0 (λ) = ρ W (λ) is the Wigner semi-circle (1.3) and ρ W,1 (λ) depends on the second moment of the diagonal entries, and moments up to and including the fourth of the offdiagonal elements [3,27]; see the recent work [19] for a convenient summary.…”

Section: Introductionmentioning

confidence: 99%

Relations between moments for the Jacobi and Cauchy random matrix ensembles

Forrester,

Rahman

2020

Preprint

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There are two versions of the Jacobi weight encountered in random matrix theory. The most common is x a (1 − x) b supported on the interval (0, 1), which is familiar from the study of the singular values of sub-blocks of unitary matrices, for example. But there is also the weight (1 − x) a (1 + x) b supported on the interval (−1, 1), which occurs for example in the study of certain random real orthogonal matrices. We show that the density for the β-ensemble with respect to the Jacobi weight supported on (−1, 1) can be related to the density for the β-ensemble with respect to the Cauchy weight (1 − ix) η (1 + ix) η by appropriate analytic continuation. This has the consequence of implying that the latter density satisfies a linear differential equation of degree three for β = 2, and of degree five for β = 1 and 4, analogues of which are already known for the Jacobi weight supported on (0, 1). We concentrate on the case a = b (Jacobi weight on (−1, 1)) and η real (Cauchy weight) since then the density is an even function and the differential equations simplify. From the differential equations, recurrences can be obtained for the moments of the Jacobi weight supported on (−1, 1) and/or the moments of the Cauchy weight. Particular attention is paid to the case β = 2 and the Jacobi weight on (−1, 1) in the symmetric case a = b, which in keeping with a recent result obtained by Assiotis et al. for the β = 2 case of the symmetric Cauchy weight (parameter η real), allows for an explicit solution of the recurrence in terms of particular continuous Hahn polynomials. Also for the symmetric Cauchy weight with η = −β(N − 1)/2 − α, after appropriately scaling α proportional to N, we use differential equations to compute terms in the 1/N 2 (1/N) expansion of the resolvent for β = 2 (β = 1, 4).

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Asymptotic correlations with corrections for the circular Jacobi β-ensemble

Forrester

Trinh

2021

Journal of Approximation Theory

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Comment on “Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices”

Cited by 4 publications

References 14 publications

Asymptotic correlations with corrections for the circular Jacobi $β$-ensemble

Asymptotic correlations with corrections for the circular Jacobi $β$-ensemble

Relations between moments for the Jacobi and Cauchy random matrix ensembles

Asymptotic correlations with corrections for the circular Jacobi β-ensemble

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