Exponential moments for disk counting statistics at the hard edge of random normal matrices

Ameur, Yacin; Charlier, Christophe; Cronvall, Joakim; Lenells, Jonatan

doi:10.4171/jst/474

Cited by 5 publications

(2 citation statements)

References 62 publications

(105 reference statements)

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“…Note that the rate of convergence (or blow-up) in (5.3) depends on Ω only through α. For β = 2, universality results on local statistics near hard edges can be found in [11,3] in the case where α = 1 and b = 1. In view of the above, new universality classes are expected for other values of α and b.…”

Section: Applicationmentioning

confidence: 94%

The multiplicative constant for the Meijer-G kernel determinant

2021

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Section: Applicationmentioning

confidence: 94%

The multiplicative constant for the Meijer-G kernel determinant

2021

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“…This physical situation is akin to the well known lowest Landau level problem of electrons in a plane and in the presence of a perpendicular magnetic field [32,45,48]. This connection with the physics of the lowest Landau levels has naturally motivated the study of the FCS in the complex Ginibre ensemble, denoted here as GinUE [1,6,7,14,25,27,28,39,73,90] (for a recent review see [16]), as well as some natural extensions of it, including the higher Landau levels [68,69,91,94], related to the socalled poly-analytic Ginibre ensemble [53]. We also refer to [78,79] and references therein for earlier work on the counting statistics of Hermitian random matrix ensembles and its applications to one-dimensional systems of trapped fermions.…”

Section: Introductionmentioning

confidence: 99%

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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Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

Ameur,

Charlier,

Cronvall

2024

J Stat Phys

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We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant $$\Gamma =2$$ Γ = 2 and that the number n of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order 1/n from the hard edge. At distances much larger than $$1/\sqrt{n}$$ 1 / n , the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the “semi-hard edge”. More precisely, we provide asymptotics for the correlation kernel $$K_{n}(z,w)$$ K n ( z , w ) as $$n\rightarrow \infty $$ n → ∞ in two microscopic regimes (with either $$|z-w| = \mathcal{O}(1/\sqrt{n})$$ | z - w | = O ( 1 / n ) or $$|z-w| = \mathcal{O}(1/n)$$ | z - w | = O ( 1 / n ) ), as well as in three macroscopic regimes (with $$|z-w| \asymp 1$$ | z - w | ≍ 1 ). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.

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Exponential moments for disk counting statistics at the hard edge of random normal matrices

Cited by 5 publications

References 62 publications

The multiplicative constant for the Meijer-G kernel determinant

The multiplicative constant for the Meijer-G kernel determinant

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

Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

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