Pseudo-symmetric random matrices: semi-Poisson and sub-Wigner statistics

Real non-symmetric matrices may have either real or complex conjugate eigenvalues. These matrices can be seen to be pseudo-symmetric as $\eta M \eta^{-1} = M^t$,wherethemetric$\eta$ could be secular (a constant matrix) or depending upon the matrix elements of $M$. Here, we construct ensembles of a large number $N$ of pseudo-symmetric $n\times n$ ($n$ large) matrices using $\mathcal{N}$ $(n(n+1)\mathrm{/}2\le \mathcal{N}\le n^2)$ independent and identically distributed (iid) random numbers as their elements. Based on our numerical calculations, we conjecture that for these ensembles the Nearest Level Spacing Distributions (NLSDs: $p(s)$) are sub-Wigner as $p_{abc}(s)=a s e^{-bs^c} (0<c <2)$andthedistributionsoftheireigenvaluesfitwellto$D(\epsilon)= A[\mbox{tanh}{(\epsilon+B)/C }-\mbox{tanh}{(\epsilon-B)/C}]$ (exceptions also discussed). These sub-Wigner NLSD are encountered in Anderson metal-insulator transition and topological transitions in a Josephson junction. Interestingly, $p(s)$ for $c=1$ is called semi-Poisson and we show that it lies close to the form $p(s)=0.59s{K}_0(0.45s^2)$ derived for the case of $2 \times 2$ pseudo-symmetric matrix where the eigenvalues are most aptly conditionally real: $E_{1,2}=a\pm \sqrt{b^2-c^2}$ which represent characteristic coalescing of eigenvalues in PT(Parity-Time)-symmetric systems.