Spectral rigidity of non-Hermitian symmetric random matrices near the Anderson transition
Yi Huang, B. I. Shklovskiǐ
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 $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ tightly bound cubic lattice, where both real and imaginary parts of onsite energies are independent random variables uniformly distributed between $\ensuremath{-}W/2$ 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 $W={W}_{c}\ensuremath{\simeq}6$ in three dimension. Here we numerically diagonalize TME $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ 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.