Bulk asymptotics for polyanalytic correlation kernels

Haimi, Antti

doi:10.1016/j.jfa.2013.11.021

Cited by 6 publications

(4 citation statements)

References 25 publications

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“…Thus, the Ginibre ensemble provides a model for the distribution of charged-like particles in the first Landau level. Until recently, there was no similar Ginibre type model for higher Landau levels, but this gap in the statistical physics literature is being filled thanks to recent work concerning the polyanalytic Ginibre ensembles [13,20,21,22,23] -see also Section 4.…”

Section: Planar Dpp's and The Ginibre Ensemblementioning

confidence: 99%

“…It has been shown by Jean Ginibre [18,19] that the distribution associated with the measure (3) is proportional to the one obtained from the N-point intensities associated with the Ginibre ensemble of dimension N. Thus, the Ginibre ensemble provides a model for the distribution of charged-like particles in the first Landau level. Until recently, there was no similar Ginibre type model for higher Landau levels, but this gap in the statistical physics literature is being filled thanks to recent work concerning the polyanalytic Ginibre ensembles [13,[20][21][22][23]-see also section 4.…”

Section: J Stat Mech (2017) 043103mentioning

confidence: 99%

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The Weyl–Heisenberg ensemble: hyperuniformity and higher Landau levels

Abreu¹,

Pereira²,

Romero³

et al. 2017

J. Stat. Mech.

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Weyl-Heisenberg ensembles are a class of determinantal point processes associated with the Schrödinger representation of the Heisenberg group. Hyperuniformity characterizes a state of matter for which (scaled) density fluctuations diminish towards zero at the largest length scales. We will prove that Weyl-Heisenberg ensembles are hyperuniform. Weyl-Heisenberg ensembles include as a special case a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels, which has recently been object of study in the realm of the Ginibre-type ensembles associated with polyanalytic functions. In addition, the family of Weyl-Heisenberg ensembles includes new structurally anisotropic processes, where point-statistics depend on the different spatial directions, and thus provide a first means to study directional hyperuniformity.PACS numbers: 05.30.-d, 05.40.-a

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Section: Planar Dpp's and The Ginibre Ensemblementioning

confidence: 99%

Section: J Stat Mech (2017) 043103mentioning

confidence: 99%

The Weyl–Heisenberg ensemble: hyperuniformity and higher Landau levels

Abreu¹,

Pereira²,

Romero³

et al. 2017

J. Stat. Mech.

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“…There the limiting kernel is not a sine kernel. In a related vein, Antti Haimi [32] established universality for weights e −nQ defined on the whole complex plane, and polyanalytic polynomials, leading to universality limits that involve associated Laguerre polynomials. Polyanalytic Ginibre ensembles were considered, for example, in [33].…”

Section: Dirichlet Polynomialsmentioning

confidence: 99%

An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

Lubinsky¹

2016

SIGMA

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Abstract. We survey the current status of universality limits for m-point correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems. The scope of the survey is quite narrow: we do not consider β ensembles for β = 2, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems.

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“…In the analytic case, the result of Rider and Virág has been generalised to more general weights by Ameur, Hedenmalm and Makarov (in [6] for test funtions with support in the bulk, and in [5] for general test functions). While it is reasonable to expect that results of the former type can be extended to polyanalytic setting by using estimates from [11], it remains an interesting question what happens with general test functions. The technique in [5] seems not to generalize to our polyanalytic case.…”

Section: Introductionmentioning

confidence: 99%

A Central limit theorem for fluctuations in Polyanalytic Ginibre ensembles

Haimi

Wennman

2017

International Mathematics Research Notices

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Abstract. We study fluctuations of linear statistics in Polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our main result is asymptotic normality of fluctuations, extending a result of Rider and Virág. As in the analytic case, the variance is composed of independent terms from the bulk and the boundary. Our methods rely on a structural formula for polyanalytic polynomial Bergman kernels which separates out the different pure q-analytic kernels corresponding to different Landau levels. The fluctuations with respect to these pure q-analytic Ginibre ensembles are also studied, and a central limit theorem is proved. The results suggest a stabilizing effect on the variance when the different Landau levels are combined together.

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Bulk asymptotics for polyanalytic correlation kernels

Abstract: For a weight function Q : C → R and a positive scaling parameter m, we study reproducing kernels K q,mQ,n of the polynomial spaces Date: November 19, 2018.

Cited by 6 publications

References 25 publications

The Weyl–Heisenberg ensemble: hyperuniformity and higher Landau levels

The Weyl–Heisenberg ensemble: hyperuniformity and higher Landau levels

An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

A Central limit theorem for fluctuations in Polyanalytic Ginibre ensembles

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