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The elliptic Ginibre ensemble: A unifying approach to local and global statistics for higher dimensions
Bielefeld Univ, Fac Phys, Bielefeld, Germany..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-7598-4521
Bielefeld Univ, Fac Math, POB 100131, D-33501 Bielefeld, Germany.;Bielefeld Univ, Fac Phys, POB 100131, D-33501 Bielefeld, Germany..
2023 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 64, no 2, article id 023503Article in journal (Refereed) Published
Abstract [en]

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows us to interpolate between the rotationally invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field at inverse temperature beta = 2. Furthermore, it represents a determinantal point process in the complex plane with the corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici [Int. J. Mod. Phys. A 11, 941 (1996)]. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a known correspondence between non-interacting fermions in a trap in d real dimensions R-d and the d-dimensional harmonic oscillator. We present a rigorous proof for the local d-dimensional bulk (sine) and edge (Airy) kernel first defined by Dean et al. [Europhys. Lett. 112, 60001 (2015)], complementing the recent results by Deleporte and Lambert [arXiv:2109.02121 (2021)]. Using the same relation to the d-dimensional harmonic oscillator in d complex dimensions C-d, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in d real and d complex dimensions. For C-d with d = 1, this corresponds to non-interacting fermions in a rotating trap.

Place, publisher, year, edition, pages
AIP Publishing , 2023. Vol. 64, no 2, article id 023503
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-325029DOI: 10.1063/5.0089789ISI: 000932383500001Scopus ID: 2-s2.0-85147829983OAI: oai:DiVA.org:kth-325029DiVA, id: diva2:1746417
Note

QC 20230328

Available from: 2023-03-28 Created: 2023-03-28 Last updated: 2023-03-28Bibliographically approved

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Duits, Maurice

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