Characteristic polynomials of non-Hermitian random band matrices

We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of a certain class of Gaussian $N\times N$ non-Hermitian random band matrices with a bandwidth $W$. Given $W,N\to \mathrm{\infty }$, we show that this behavior near the point in the bulk of the spectrum exhibits the crossover at $W\sim \sqrt{N}$: it coincides with those for Ginibre ensemble for $W\gg \sqrt{N}$, and factorized as $1\ll W\ll \sqrt{N}$. The result is the first step toward the proof of Anderson’s type transition for non-Hermitian random band matrices.