On the number of real eigenvalues of a product of truncated orthogonal random matrices

Little, Alex; Mezzadri, Francesco; Simm, Nick

doi:10.1214/21-ejp732

Cited by 8 publications

(10 citation statements)

References 39 publications

(73 reference statements)

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“…Let us also mention that similar formulas can be found in the context of induced Ginibre, spherical Ginibre, and truncated orthogonal matrices, see e.g. [17,18,20,23,25,38,40,46] and also [9,Section 4] for a comprehensive review.…”

Section: Proofs Of Main Resultsmentioning

confidence: 89%

Real Eigenvalues of Elliptic Random Matrices

Byun

Kang

Lee

2021

International Mathematics Research Notices

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We consider the real eigenvalues of an $(N \times N)$ real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $\tau _N\in [0,1]$. In the almost-Hermitian regime where $1-\tau _N=\Theta (N^{-1})$, we obtain the large-$N$ expansion of the mean and the variance of the number of the real eigenvalues. Furthermore, we derive the limiting densities of the real eigenvalues, which interpolate the Wigner semicircle law and the uniform distribution, the restriction of the elliptic law on the real axis. Our proofs are based on the skew-orthogonal polynomial representation of the correlation kernel due to Forrester and Nagao.

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Section: Proofs Of Main Resultsmentioning

confidence: 89%

Real Eigenvalues of Elliptic Random Matrices

Byun

Kang

Lee

2021

International Mathematics Research Notices

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“…The methods developed in this paper likely apply to various other models of real asymmetric random matrices. One particular example is a product of truncated Haar distributed orthogonal random matrices, which has been the subject of recent investigations [FK18,FIK20,LMS21]. As in the case of real Ginibre random matrices, eigenvalue statistics of such products are again described by a Pfaffian point process.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…These quantities all have convenient integral representations, and the following three propositions follow from applying the Laplace method of asymptotics to those integral representations. We do not give the details of the proof here, but instead refer the author to Appendix A of [LMS21] where the main ideas are discussed. These asymptotics were also considered in the context of complex Ginibre random matrices in [AB12].…”

Section: Preliminaries and Proof Of The Global Approximationmentioning

confidence: 99%

“…Note that in this regime the second term in (2.2) is exponentially small due to the factor x N−1 and can be neglected. For the first term in (2.2) we consider a large constant M > 0 and consider the contribution from the interval x ∈ [0, M N − The following Lemma is given in [LMS21]. For completeness we repeat the proof here.…”

Section: Appendix B Uniform Asymptotics Of a Truncated Hypergeometric...mentioning

confidence: 99%

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Fluctuations and correlations for products of real asymmetric random matrices

FitzGerald,

Simm

2021

Preprint

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We study the real eigenvalue statistics of products of independent real Ginibre random matrices. These are matrices all of whose entries are real i.i.d. standard Gaussian random variables. For such product ensembles, we demonstrate the asymptotic normality of suitably normalised linear statistics of the real eigenvalues and compute the limiting variance explicitly in both global and mesoscopic regimes. A key part of our proof establishes uniform decorrelation estimates for the related Pfaffian point process, thereby allowing us to exploit weak dependence of the real eigenvalues to give simple and quick proofs of the central limit theorems under quite general conditions. We also establish the universality of these point processes. We compute the asymptotic limit of all correlation functions of the real eigenvalues in the bulk, origin and spectral edge regimes. By a suitable strengthening of the convergence at the edge, we also obtain the limiting fluctuations of the largest real eigenvalue. Near the origin we find new limiting distributions characterising the smallest positive real eigenvalue.

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“…The products of M truncated orthogonal matrices in §4.4 have also been studied in the literature [88,87,129]. In particular, in the strong non-orthogonality, it was shown in [129] that the leading order asymptotic of the expected number of real eigenvalues is of the form (4.32) multiplied by √ M; cf. (4.48).…”

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confidence: 99%

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Byun¹,

Forrester²

2023

Preprint

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This is part II of a review relating to the three classes of random non-Hermitian Gaussian matrices introduced by Ginibre in 1965. While part I restricted attention to the GinUE (Ginibre unitary ensemble) case of complex elements, in this part the cases of real elements (GinOE, denoting Ginibre orthogonal ensemble) and quaternion elements represented as 2 × 2 complex blocks (GinSE, denoting Ginibre symplectic ensemble) are considered. The eigenvalues of both GinOE and GinSE form Pfaffian point processes, which are more complicated than the determinantal point processes resulting from GinUE. Nevertheless, many of the obstacles that have slowed progress on the development of traditional aspects of the theory have now been overcome, while new theoretical aspects and new applications have been identified. This permits a comprehensive account of themes addressed too in the complex case: eigenvalue probability density functions and correlation functions, limit formulas for correlation functions, fluctuation formulas, sum rules, gap probabilities and eigenvector statistics, among others. Distinct from the complex case is the need to develop a theory of skew orthogonal polynomials corresponding to the skew inner product associated with the Pfaffian. Another distinct theme is the statistics of real eigenvalues, which is unique to GinOE. These appear in a number of applications of the theory, coming from areas as diverse as diffusion processes and persistence in statistical physics, topologically driven parametric energy level crossings for certain quantum dots, and equilibria counting for a system of random nonlinear differential equations.

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On the number of real eigenvalues of a product of truncated orthogonal random matrices

Cited by 8 publications

References 39 publications

Real Eigenvalues of Elliptic Random Matrices

Real Eigenvalues of Elliptic Random Matrices

Fluctuations and correlations for products of real asymmetric random matrices

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

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