On the almost-circular symplectic induced Ginibre ensemble

Byun, Sung‐Soo; Charlier, Christophe

doi:10.48550/arxiv.2206.06021

Cited by 4 publications

(4 citation statements)

References 37 publications

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“…From the general universality principle of random matrix theory, it can be expected that for any fixed τ ∈ [0, 1), the local statistics of the ensemble in the large system coincide with the one (equation (1.5)) obtained from the case τ = 0. Based on the skew-orthogonal Hermite polynomials introduced by Kanzieper [41], such a statement was recently proved in [7] for p = 0 and in [22] for general p ∈ R. (We mention that the local statistic at p = 0 can be special for certain random matrix models [8,21,23,31,40]. )…”

Section: Symplectic Elliptic Ginibre Ensemble In the Almost-hermitian...mentioning

confidence: 95%

Wronskian structures of planar symplectic ensembles

2022

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We consider the eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which are known to form a Pfaffian point process 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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Section: Symplectic Elliptic Ginibre Ensemble In the Almost-hermitian...mentioning

confidence: 95%

Wronskian structures of planar symplectic ensembles

2022

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“…This is because the corresponding eigenvalues of the Fredholm determinant (or Pfaffian) are explicitly known, and we refer to [6] for a comprehensive study in complex and symplectic ensembles. (See also [15,18,32] and references therein for recent development in this direction.) Other quantities have been analysed as well, including the number of eigenvalues in a generic, non-rotational invariant domain [1,2,48] or the probability of overcrowding a domain [5,7,42].…”

Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 99%

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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“…This is because the corresponding eigenvalues of the Fredholm determinant (or Pfaffian) are explicitly known, and we refer to [8] for a comprehensive study in complex and symplectic ensembles. (See also [15,19,32] and references therein for recent development in this direction.) Other quantities have been analysed as well, including the number of eigenvalues in a generic, non-rotational invariant domain [1,2,47] or the probability of overcrowding a domain [7,9,42].…”

Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 99%

Universality of the number variance in rotational invariant two-dimensional Coulomb gases

Akemann¹,

Byun²,

Ebke³

2022

Preprint

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An exact map was established by Lacroix-A-Chez-Toine, Majumdar, and Schehr in [44] 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 Na in a disc of radius a. Extending the work [44] 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 Na is universal in the large-N limit, when measured in units of the mean density proportional to ∆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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On the almost-circular symplectic induced Ginibre ensemble

Cited by 4 publications

References 37 publications

Wronskian structures of planar symplectic ensembles

Wronskian structures of planar symplectic ensembles

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

Universality of the number variance in rotational invariant two-dimensional Coulomb gases

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