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

Byun, Sung‐Soo; Forrester, Peter J.

doi:10.48550/arxiv.2301.05022

Cited by 8 publications

(17 citation statements)

References 136 publications

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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: 85%

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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: 85%

“…Nowadays these models are called the real, complex and quaternion Ginibre ensembles (denoted as GinOE, GinUE and GinSE), and they are well-studied in the past half century. For a recent review on the Ginibre ensembles, we refer to the papers [8,9].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Real Eigenvalues of Elliptic Random Matrices

Byun

Kang

Lee

2021

International Mathematics Research Notices

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“…Also here, the integrable structure (2.7) remains, with the corresponding skew-kernel (2.8) given in terms of planar skew-orthogonal polynomials, cf. [17].…”

Section: Full Counting Statistics Of Planar Symplectic Ensemblesmentioning

confidence: 99%

“…Second, we provide the origin scaling of all three types of correlation functions as they will be needed in the computation of the number variance of the GinOE in this limit. For more details we refer to [17] as well as to the original literature.…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

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Universality of the Number Variance in Rotational Invariant Two-Dimensional Coulomb Gases

2022

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An exact map was established by Lacroix-A-Chez-Toine et al. in (Phys Rev A 99(2):021602, 2019) between the N complex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble, and the positions of N non-interacting Fermions in a rotating trap in the ground state. An important quantity is the statistics of the number of Fermions $$\mathcal {N}_a$$ N a in a disc of radius a. Extending the work (Lacroix-A-Chez-Toine et al., in Phys Rev A 99(2):021602, 2019) covering Gaussian and rotationally invariant potentials Q, we present a rigorous analysis in planar complex and symplectic ensembles, which both represent 2D Coulomb gases. We show that the variance of $$\mathcal {N}_a$$ N a is universal in the large-N limit, when measured in units of the mean density proportional to $$\Delta Q$$ Δ Q , which itself is non-universal. This holds in the large-N limit in the bulk and at the edge, when a finite fraction or almost all Fermions are inside the disc. In contrast, at the origin, when few eigenvalues are contained, it is the singularity of the potential that determines the universality class. We present three explicit examples from the Mittag-Leffler ensemble, products of Ginibre matrices, and truncated unitary random matrices. Our proofs exploit the integrable structure of the underlying determinantal respectively Pfaffian point processes and a simple representation of the variance in terms of truncated moments at finite-N.

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Spectral moments of the real Ginibre ensemble

Byun,

Forrester

2024

Ramanujan J

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The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large N expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments $$M_{2p}^\textrm{r}$$ M 2 p r . The latter are expressed in terms of the $${}_3 F_2$$ 3 F 2 hypergeometric functions, with a simplification to the $${}_2 F_1$$ 2 F 1 hypergeometric function possible for $$p=0$$ p = 0 and $$p=1$$ p = 1 , allowing for the large N expansion of these moments to be obtained. The large N expansion involves both integer and half-integer powers of 1/N. The three-term recurrence then provides the large N expansion of the full sequence $$\{ M_{2p}^\textrm{r} \}_{p=0}^\infty $$ { M 2 p r } p = 0 ∞ . Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large N expansion of these quantities are determined.

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

Cited by 8 publications

References 136 publications

Real Eigenvalues of Elliptic Random Matrices

Real Eigenvalues of Elliptic Random Matrices

Universality of the Number Variance in Rotational Invariant Two-Dimensional Coulomb Gases

Spectral moments of the real Ginibre ensemble

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