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

Abstract

We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris and Economou (TME). This is a $L\times L \times L$ tightly bound cubic lattice, where both real and imaginary parts of on-site 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 $W= W_c \simeq 6$ in three dimension. Here we numerically diagonalize TME $L \times L \times L$ 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 $N_c(L,W)$ eigenvalues. We find that $N_c(L,W)$ is proportional to $L$ and grows with decreasing $W$ similarly to the number of energy levels $N_c$ in the Thouless energy band of the Anderson model.