On the almost‐circular symplectic induced Ginibre ensemble

Byun, Sung‐Soo; Charlier, Christophe

doi:10.1111/sapm.12537

Cited by 8 publications

(8 citation statements)

References 46 publications

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“…Another interesting double scaling limit arises in the so-called almost-circular regime, where the spectrum tends to form a thin annulus of width O(1/N). In this case, the scaling limits both at the real axis as well as away from the real axis were obtained in [34]. For the former case, the limiting correlation functions are Pfaffians, which interpolates the bulk scaling limits of the GinSE and of the antisymmetric Gaussian Hermitian ensemble.…”

Section: Induced Ginsementioning

confidence: 76%

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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Section: Induced Ginsementioning

confidence: 76%

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

Byun¹,

Forrester²

2023

Preprint

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“…Very recently, the scaling limits of the symplectic induced Ginibre ensemble in the almost-circular (or weakly non-unitary) regime were obtained in [18]. While the almost-Hermitian [5,39,40] and almost-circular [9,23] ensembles have the same bulk scaling limits in the complex symmetry class, those are different in the symplectic symmetry class in the vicinity of the real line due to the lack of the translation invariance; see [18] for further details.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Nevertheless, the crux of Proposition 1.1 is the transforms (1.8) and (1.9) in the expression (1.7), which lead to a simple differential equation (1.11) stated in Proposition 1.1 (b). To be more concrete, let us mention that in general, one strategy for performing an asymptotic analysis on a double summation appearing in the skew-orthogonal polynomial kernel is to derive a "proper" differential equation satisfied by the kernel; see [2,4,18,19,45]. (Such a differential equation for the two-dimensional ensemble is broadly called the generalised Christoffel-Darboux formula [4,19,52]).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Spherical Induced Ensembles with Symplectic Symmetry

Byun¹,

Forrester²

2023

SIGMA

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We consider the complex eigenvalues of the induced spherical Ginibre ensemble with symplectic symmetry and establish the local universality of these point processes along the real axis. We derive scaling limits of all correlation functions at regular points both in the strong and weak non-unitary regimes as well as at the origin having spectral singularity. A key ingredient of our proof is a derivation of a differential equation satisfied by the correlation kernels of the associated Pfaffian point processes, thereby allowing us to perform asymptotic analysis.

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“…Furthermore, for the GinUE, more precise asymptotic behaviour is available in [26], see also remark 2.15. We mention that the order O( √ N) is closely related to the order of the semilarge gap probabilities, see [15,27]. Let us also note that for the GinUE case, the optimal convergence rate of O(N −1/2 ) to the circular law was established in [51].…”

Section: 23)mentioning

confidence: 90%

Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble

Akemann,

Byun,

Ebke

et al. 2023

J. Phys. A: Math. Theor.

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In this article, we compute and compare the statistics of the number of eigenvalues in a centred disc of radius R in all three Ginibre ensembles. We determine the mean and variance as functions of R in the vicinity of the origin, where the real and symplectic ensembles exhibit respectively an additional attraction to or repulsion from the real axis, leading to different results. In the large radius limit, all three ensembles coincide and display a universal bulk behaviour of O ( R 2 ) for the mean, and O(R) for the variance. We present detailed conjectures for the bulk and edge scaling behaviours of the real Ginibre ensemble, having real and complex eigenvalues. For the symplectic ensemble we can go beyond the Gaussian case (corresponding to the Ginibre ensemble) and prove the universality of the full counting statistics both in the bulk and at the edge of the spectrum for rotationally invariant potentials, extending a recent work which considered the mean and the variance. This statistical behaviour coincides with the universality class of the complex Ginibre ensemble, which has been shown to be associated with the ground state of non-interacting fermions in a two-dimensional rotating harmonic trap. All our analytical results and conjectures are corroborated by numerical simulations.

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On the almost‐circular symplectic induced Ginibre ensemble

Cited by 8 publications

References 46 publications

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

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

Spherical Induced Ensembles with Symplectic Symmetry

Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble

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