Pseudosymmetric random matrices: Semi-Poisson and sub-Wigner statistics.

Real nonsymmetric matrices may have either real or complex conjugate eigenvalues. These matrices can be seen to be pseudosymmetric as ηMη^{-1}=M^{t}, where the metric η could be secular (a constant matrix) or depending upon the matrix elements of M. Here we construct ensembles of a large number N of pseudosymmetric n×n (n large) matrices using N[n(n+1)/2≤N≤n^{2}] independent and identically distributed 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)=ase^{-bs^{c}}(0<c<2) and the distributions of their eigenvalues fit well to D(ε)=A[tanh{(ε+B)/C}-tanh{(ε-B)/C}] (exceptions also discussed). These sub-Wigner NLSDs 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)=0.59sK_{0}(0.45s^{2}) derived for the case of 2×2 pseudosymmetric matrix where the eigenvalues are most aptly conditionally real, E_{1,2}=a±sqrt[b^{2}-c^{2}], which represent characteristic coalescing of eigenvalues in parity-time (PT) -symmetric systems.