Rate of convergence for products of independent non-Hermitian random matrices

Jalowy, Jonas

doi:10.1214/21-ejp625

Cited by 4 publications

(1 citation statement)

References 31 publications

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“…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]. This result was further generalised to the products of GinUE matrices as well [58].…”

Section: 1mentioning

confidence: 67%

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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Section: 1mentioning

confidence: 67%

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

2022

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Quantitative results for banded Toeplitz matrices subject to random and deterministic perturbations

O’Rourke

Wood

2023

Linear Algebra and its Applications

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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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Rate of convergence for products of independent non-Hermitian random matrices

Cited by 4 publications

References 31 publications

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

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

Quantitative results for banded Toeplitz matrices subject to random and deterministic perturbations

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

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