Real eigenvalues of elliptic random matrices

Byun, Sung-Soo; Kang, Nam-Gyu; Lee, Ji Oon; Lee, Jinyeop

doi:10.48550/arxiv.2105.11110

Cited by 4 publications

(5 citation statements)

References 35 publications

(47 reference statements)

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“…In the critical regime (1.2), the situation becomes somewhat similar to the real elliptic Ginibre ensemble in the weakly asymmetric regime [8], in the sense that a non-trivial portion of the N eigenvalues is real, cf. [6,16].…”

Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 99%

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The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

Akemann¹,

Byun²

2022

Preprint

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We study the product Pm of m real Ginibre matrices with Gaussian elements of size N , which has received renewed interest recently. Its eigenvalues, which are either real or come in complex conjugate pairs, become all real with probability one when m → ∞ at fixed N . In this regime the statistics becomes deterministic and the Lyapunov spectrum has been derived long ago. On the other hand, when N → ∞ and m is fixed, it can be expected that away from the origin the same local statistics as for a single real Ginibre ensemble at m = 1 prevails. Inspired by analogous findings for products of complex Ginibre matrices, we introduce a critical scaling regime when the two parameters are proportional, m = αN . We derive the expected number, variance and rescaled density of real eigenvalues in this critical regime. This allows us to interpolate between previous recent results in the above mentioned limits when α → ∞ and α → 0, respectively.

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Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 99%

“…We also refer the reader to [6] for the use of such an identity in the context of the real elliptic Ginibre ensemble. Notice that this expression is very much reminiscent to the result in determinantal point processes with a single scalar kernel.…”

Section: Let Us Begin With Deriving An Alternative Representation Of ...mentioning

confidence: 99%

The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

Akemann¹,

Byun²

2022

Preprint

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“…As it allows to perform a suitable asymptotic analysis, this formula has turned out to be very useful in various situations, see e.g. [8,13,32]. In a similar spirit, we emphasise that Proposition 1.1 can also be applied to other cases including the almost-Hermitian regime.…”

Section: Figure 1 Eigenvalues Of the Elliptic Ginibre Ensemblementioning

confidence: 95%

Universal scaling limits of the symplectic elliptic Ginibre ensemble

Byun,

Ebke

2021

Preprint

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We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behaviour of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel-Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.

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“…Invoking this asymptotics, the resulting oscillatory integral in (4.4) can be analysed in the same way as [21,Lemma 4.4]. To be more precise, it has the asymptotic form √ N Re p 0 a(t)e iN φ(t) dt, where 4.3.…”

Section: Now Lemma 43 Completes the Proofmentioning

confidence: 99%

Wronskian structures of planar symplectic ensembles

Byun¹,

Ebke²,

Seo³

2021

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We consider the complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which is known to form Pfaffian point processes in the plane. It was recently discovered that the limiting correlation kernel of the symplectic Ginibre ensemble in the vicinity of the real line can be expressed in a unified form of a Wronskian. We derive scaling limits for variations of the symplectic Ginibre ensemble and obtain such Wronskian structures for the associated universality classes. These include almost-Hermitian bulk/edge scaling limits of the elliptic symplectic Ginibre ensemble and edge scaling limits of the symplectic Ginibre ensemble with boundary confinement. Our proofs follow from the generalised Christoffel-Darboux formula for the former and from the Laplace method for the latter. Based on such a unified integrable structure of Wronskian form, we also provide an intimate relation between the function in the argument of the Wronskian in the symplectic symmetry class and the kernel in the complex symmetry class which form determinantal point processes in the plane.

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Real eigenvalues of elliptic random matrices

Cited by 4 publications

References 35 publications

The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

Universal scaling limits of the symplectic elliptic Ginibre ensemble

Wronskian structures of planar symplectic ensembles

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