Spectral statistics for ensembles of various real random matrices

Abstract

We investigate spacing statistics $p(s)$ and distribution of eigenvalues $D(\hat{I}\mu )$ for ensembles of various real random matrices (of order $n\times n,n=2$ and $n>>2$) where the matrix-elements have various Probability Distribution Function (PDF: $f(x)$) including Gaussian. We construct ensembles of $1000$, $100\times 100$ real random matrices $R$, $C$ (cyclic) and $T$ (tridiagonal) and real symmetric matrices: ${\mathcal{R}}^{\prime }$, $\mathcal{R}=R+R^t$, $\mathcal{Q}=R{R}^t$, $\mathcal{C}$ (cyclic), $\mathcal{T}$ (tridiagonal), $T^{\prime }$ (pseudo-symmetric Tridiagonal), $\hat{I}˜$ (Toeplitz) , $\mathcal{D}=C{C}^t$ and $\mathcal{S}=T{T}^t$. We find that the spacing distribution of the adjacent levels of matrices $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$ under any symmetric PDF of matrix elements is $p_{AB}(s)=As{e}^{-B{s}^2}$ which approximately conforms to the Wigner surmise as $A/2\approx B\approx \ddot{I}€/4$. But under asymmetric PDFs we observe $A/2\approx B>>\ddot{I}€/4$, where $A,B$ are also sensitive to the choice of the matrix and the PDF. More interestingly, the real symmetric matrices $\mathcal{C},\mathcal{T},\mathcal{Q}$, $\hat{I}˜$ (excepting $\mathcal{D}$ and $\mathcal{S}$) and $T^{\prime }$ (pseudo-symmetric tridiagonal) all conform to the Poisson distribution $p_{\hat{I}}¼(s)=\hat{I}¼e^{-\hat{I}¼s}$, where $\hat{I}¼$ depends upon the choice of the matrix and PDF. Let complex eigenvalues of $R$, $C$ and $T$ be $E_n^c$. We show that all $p(s)$ arising due to $\Re (E_n^c)$, $\Im (E_n^c)$ and $|E_n^c|$ of $R$, $C$ and $T$ are also of Poisson type: $\hat{I}¼e^{-\hat{I}¼s}$. We observe $p(s)$ as half-Gaussian for two real eigenvalues of $C$. For real matrices $R,C,T$, we associate new types of $p(s)$ with them. Lastly, we study the distribution $D(\hat{I}\mu )$ of eigenvalues of symmetric matrices (of large order) discussed above.

12 pages, 11 figures and 4 tables