Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at noninteger β

Gernot Akemann, Adam Mielke, and Patricia Päßler
Phys. Rev. E 106, 014146 – Published 28 July 2022

Abstract

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature β. For 2×2 matrices with Gaussian distribution we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective βeff(β) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of β. It reproduces the 2D Poisson distribution at β=0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values β=1.4 and β=2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.

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  • Received 5 January 2022
  • Revised 21 April 2022
  • Accepted 7 July 2022

DOI:https://doi.org/10.1103/PhysRevE.106.014146

©2022 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Gernot Akemann1,*, Adam Mielke2,†, and Patricia Päßler1,‡

  • 1Faculty of Physics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany
  • 2Technical University of Denmark, Asmussens Allé, Building 303B, 2800 Kgs. Lyngby, Denmark

  • *akemann@physik.uni-bielefeld.de
  • admi@dtu.dk
  • patricia@physik.uni-bielefeld.de

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Vol. 106, Iss. 1 — July 2022

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