Characteristic polynomials of non-Hermitian random band matrices near the threshold

The paper arXiv:2510.04255 shows that the asymptotic behavior of the second correlation function of characteristic polynomials of the $N\times N$ non-Hermitian random band matrices with a bandwidth $W$ exhibits the transition at $W\sim \sqrt{N}$, as $W,N\to \mathrm{\infty }$: it coincides with those for Ginibre ensemble for $W\gg \sqrt{N}$, and factorized as $1\ll W\ll \sqrt{N}$. In this work we extend the techniques of arXiv:2510.04255 to study the critical regime when the bandwidth $W$ is proportional to the threshold $\sqrt{N}$.