2008/06/17 by Hans-Jürgen Sommers, Waldemar Wieczorek · 45 citations
Computer Science · Mathematics · Physics and Astronomy · #Eigenvalues and eigenvectors #Gaussian #Grassmannian #Invariant (physics) #Kernel (algebra) #Mathematical functions and polynomials #Matrix Theory and Algorithms #Pfaffian #Random Matrices and Applications #Simple (philosophy) #Symplectic geometry #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8113/41/40/405003
published in Journal of Physics A Mathematical and Theoretical 41(40), 405003 (Institute of Physics) · 25 pages, 7 figures
arxiv created 2008/06/17 · openalex publication_date 2008/09/11 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We rederive in a simplified version the Lehmann-Sommers eigenvalue distribution for the Gaussian ensemble of asymmetric real matrices, invariant under real orthogonal transformations, as a basis for a detailed derivation of a Pfaffian generating functional for n-point densities. This produces a simple free-fermion diagram expansion for the correlations leading to quaternion determinants in each order n. All will explicitly be given with the help of a very simple symplectic kernel for even dimension N. The kernel is valid both for complex and real eigenvalues and describes a deep connection between both. A slight modification by an artificial additional Grassmannian solves also the more complicated odd-N case. As illustration we present some numerical results in the space ℂn of complex eigenvalue n-tuples.