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Eigenvalue statistics of random real matrices. (English) Zbl 0990.82528
Summary: Completing Ginibre’s work we determine the joint probability density of eigenvalues in a Gaussian ensemble of real asymmetric matrices, which is invariant under orthogonal transformations. The symmetry parameter tau may vary from \(-1\) (antisymmetric ensemble) through 0 (completely asymmetric ensemble) to \(+1\) (symmetric ensemble). The elliptic law for the average density of eigenvalues in the limit of large dimension is recovered. Matrices of the type considered appear in models for neural-network dynamics and dissipative quantum dynamics.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
15A18 Eigenvalues, singular values, and eigenvectors
15B52 Random matrices (algebraic aspects)
82C32 Neural nets applied to problems in time-dependent statistical mechanics
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