Singular Values of Products of Ginibre Random Matrices

Witte, N. S.; Forrester, Peter J.

doi:10.1111/sapm.12147

Cited by 15 publications

(17 citation statements)

References 29 publications

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“…which is consistent with a recent result obtained by Witte and Forrester in [42, Corollary 3.1]. Furthermore, in the special case ν 1 = − 1 2 and ν 2 = 0, the sub-leading coefficient is equal to b (1) = − 3 2 5/3 , which agrees with formula (3.82) from the same paper [42]. The same agreement is achieved for j = 3, α = 0, θ = 2 after the identification s → 2 √ s (see [42, formula (3.79)]), the coefficients being a (3) = 9 2 11/3 and b (3) = − 3 2 7/3 .…”

Section: )supporting

confidence: 92%

Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

Claeys

Girotti

Stivigny

2017

International Mathematics Research Notices

134

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We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer G-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a 2×2 Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics.

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Section: )supporting

confidence: 92%

Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

Claeys

Girotti

Stivigny

2017

International Mathematics Research Notices

134

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“…Less well understood is the nonlinear differential system implied by the correlation kernel based on these special functions. These are relevant to the study of gap probabilities; see [WF17,MF18].…”

Section: Introductionmentioning

confidence: 99%

Schwinger–Dyson and loop equations for a product of square Ginibre random matrices

Dartois

Forrester

2020

J. Phys. A: Math. Theor.

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In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e. the Stieltjes transform of the correlation functions). We obtain using Schwinger-Dyson equation (SDE) techniques the general loop equations satisfied by the resolvents. In order to deal with the product structure of the random matrix of interest, we consider SDEs involving the integral of higher derivatives. One of the advantage of this technique is that it bypasses the reformulation of the problem in terms of singular values. As a byproduct of this study we obtain the large N limit of the Stieltjes transform of the 2-point correlation function, as well as the first correction to the Stieltjes transform of the density, giving us access to corrections to the smoothed density. In order to pave the way for the establishment of a topological recursion formula we also study the geometry of the corresponding spectral curve. This paper also contains explicit results for different resolvents and their corrections.

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“…[16, Table 3] for a list including references, for the product of r ≥ 2 independent matrices the corresponding Painlevé type systems of equations [40] become very rapidly cumbersome, cf. [43]. This is the reason why we will focus on the kernel instead.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

Akemann¹,

Checinski²,

Liu³

et al. 2019

Ann. Inst. H. Poincaré Probab. Statist.

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We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer G-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.

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Singular Values of Products of Ginibre Random Matrices

Cited by 15 publications

References 29 publications

Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

Schwinger–Dyson and loop equations for a product of square Ginibre random matrices

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

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