Analogies between random matrix ensembles and the one-component plasma in two-dimensions

Abstract

The eigenvalue PDF for some well known classes of non-Hermitian random matrices --- the complex Ginibre ensemble for example --- can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We address this theme in a systematic fashion, identifying the plasma system for the Ginibre ensemble of non-Hermitian Gaussian random matrices $G$, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix $G_1^{-1}{G}_2$, and the ensemble formed by truncating unitary matrices, as well as for products of such matrices. We do this when each has either real, complex or real quaternion elements. One consequence of this analogy is that the leading form of the eigenvalue density follows as a corollary. Another is that the eigenvalue correlations must obey sum rules known to characterise the plasma system, and this leads us to a exhibit an integral identity satisfied by the two-particle correlation for real quaternion matrices in the neighbourhood of the real axis. Further random matrix ensembles investigated from this viewpoint are self dual non-Hermitian matrices, in which a previous study has related to the one-component plasma system in a disk at inverse temperature $\beta =4$, and the ensemble formed by the single row and column of quaternion elements from a member of the circular symplectic ensemble.

25 pages

References in corpus (9)

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• Truncations of Random Orthogonal Matrices
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Cited by in corpus (23)

• Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos
• Universality classes of non-Hermitian random matrices
• Large deviations of radial statistics in the two-dimensional one-component plasma
• The high temperature crossover for general 2D Coulomb gases
• The boundary density profile of a Coulomb droplet. Freezing at the edge
• Realizable hyperuniform and nonhyperuniform particle configurations with targeted spectral functions via effective pair interactions
• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Symmetries of the Quaternionic Ginibre Ensemble
• Simulating Coulomb gases and log-gases with hybrid Monte Carlo algorithms
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality
• Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices
• Repulsion in low temperature $\beta$-ensembles
• Scaling Limits of Planar Symplectic Ensembles
• Spherical Induced Ensembles with Symplectic Symmetry
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Analytic approach for the number statistics of non-Hermitian random matrices
• Truncations of random symplectic unitary matrices
• Two-Dimensional Coulomb Gas on an Elliptic Annulus
• Two dimensional Coulomb gas in a non-conservative trap