Spacing distribution in the 2D Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at non-integer $\beta$

Abstract

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature $\beta$. For $2\times 2$ matrices with Gaussian distribution we analytically compute the nearest neighbour 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 ${\beta }_{\mathrm{eff}}(\beta )$ in our analytic formula, that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of $\beta$. It reproduces the 2D Poisson distribution at $\beta =0$ 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 non-integer values $\beta =1.4$ and $\beta =2.6$ 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.

arXiv comment6+9 pages, 4+5 figures; v2: 3 references added, typos corrected; v3: version to appear in Phys. Rev. E, 12 pages

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Cited by in corpus (12)

• Diagnosing non-Hermitian Many-Body Localization and Quantum Chaos via Singular Value Decomposition
• Replica Symmetry Breaking for the Integrable Two-Site Sachdev-Ye-Kitaev Model
• Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking
• Interactions Between Different Birds of Prey as a Random Point Process
• Correspondence principle, dissipation, and Ginibre ensemble
• Electrodynamics of Vortices in Quasi-2D Scalar Bose-Einstein Condensates
• PT -symmetry breaking and universal spectral statistics in quantum kicked rotors
• Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics
• Surmise for random matrices' level spacing distributions beyond nearest-neighbors
• A class of 2x2 correlated random-matrix models with Brody spacing distribution
• A matrix model of a non-Hermitian $\beta$-ensemble
• Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI${}^{†}$ and AII${}^{†}$