2010/12/31 by Yogesh N. Joglekar, William A. Karr
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics #Diagonal #Distribution (mathematics) #Eigenvalues and eigenvectors #Gaussian #Geometry #Hermitian matrix #Hilbert space #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Physics #Product (mathematics) #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Self-adjoint operator #Sigma #Unitary matrix #Unitary state #Zero (linguistics) #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.83.031122
published as Phys. Rev. E 83, 031122 (2011) · 5 pages, 5 figures, revised text
arxiv created 2011/02/09 · openalex publication_date 2011/03/18 · arxiv updated 2011/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We investigate the level density σ(x) and the level-spacing distribution p(s) of random matrices M = AF ≠ M†, where F is a (diagonal) inner product and A is a random, real, symmetric or complex, Hermitian matrix with independent entries drawn from a probability distribution q(x) with zero mean and finite higher moments. Although not Hermitian, the matrix M is self-adjoint with respect to F and thus has purely real eigenvalues. We find that the level density σF(x) is independent of the underlying distribution q(x) and solely characterized by F, and therefore generalizes the Wigner semicircle distribution σW(x). We find that the level-spacing distributions p(s) are independent of q(x), and are dependent upon both the inner product F and whether A is real or complex, and therefore generalize the Wigner surmise for level spacing. Our results suggest F-dependent generalizations of the well-known Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble classes.