Spectral rigidity of non-Hermitian symmetric random matrices near the Anderson transition

We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris, and Economou (TME). This is a L×L×L tightly bound cubic lattice, where both real and imaginary parts of onsite energies are independent random variables uniformly distributed between-W/2 and W/2. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at W=Wc≃6 in three dimension. Here we numerically diagonalize TME L×L×L cubic lattice matrices and calculate the number variance of eigenvalues in a disk of their complex plane. We show that on the metallic side W<6 of the Anderson transition, complex eigenvalues repel each other as strongly as in the complex Ginibre ensemble only in a disk containing Nc(L,W) eigenvalues. We find that Nc(L,W) is proportional to L and grows with decreasing W similarly to the number of energy levels Nc in the Thouless energy band of the Anderson model.