Spectral statistics for ensembles of various real random matrices

We investigate spacing statistics for ensembles of various real random matrices where the matrixelements have various Probability Distribution Function (PDF: f(x)) including Gaussian. For two modifications of 2 × 2 matrices with various PDFs, we derived that spacing distribution p(s) of adjacent energy eigenvalues are distinct. Nevertheless, they show the linear level repulsion near s = 0 as αs where α depends on the choice of the PDF. We also derive the distribution of eigenvalues D( ) for these matrices. We construct ensembles of 1000, 100 × 100 real random matrices R, C (cyclic) and T (tridiagonal) and real symmetric matrices: R′, R = R+R, Q = RR, C (cyclic), T (tridiagonal), T ′ (pseudo-symmetric Tridiagonal), Θ (Toeplitz) , D = CC and S = TT . We find that the spacing distribution of the adjacent levels of matrices R and R′ under any symmetric PDF of matrix elements is pAB(s) = Ase −Bs which approximately conforms to the Wigner surmise as A/2 ≈ B ≈ π/4. But under asymmetric PDFs we observe A/2 ≈ B >> π/4, where A,B are also sensitive to the choice of the matrix and the PDF. More interestingly, the real symmetric matrices C, T ,Q, Θ (excepting D and S) and T ′ (pseudo-symmetric tridiagonal) all conform to the Poisson distribution pμ(s) = μe −μs, where μ depends upon the choice of the matrix and PDF. Let complex eigenvalues of R, C and T be E n. We show that all p(s) arising due to <(E n), =(E n) and |E n| of R, C and T are also of Poisson type: μe−μs. So far the spacing distribution of mixed eigenvalues of an ensemble of real symmetric random matrix is known to be of Poisson type. We observe p(s) as half-Gaussian for two real eigenvalues of C. Defining spacing as Sk = |E k+1 −E k| for real matrices R,C, T ; when the complex eigenvalues are ordered by their real parts, we associate new types of p(s) with them. Lastly, we study the distribution D( ) of eigenvalues of symmetric matrices (of large order) discussed above. For R and R′, we recover semi-circle law and for C we get the known form. Θ gives a Gaussian and, T and T ′ yield super-Gaussian distributions. For the secondary matrices Q,D,S; D( ) turns out to be exponential/sub-exponential.