Characteristic polynomials of non-Hermitian random band matrices

TL;DR

The paper analyzes the second correlation function of the characteristic polynomials for Gaussian non-Hermitian random band matrices with bandwidth $W$, in the limit $N,W\to\infty$. Employing a SUSY transfer-matrix framework, it rewrites $\Theta$ as the $N-1$ power of a transfer operator $\mathcal{K}_{\zeta}$ acting on $2\times2$ matrices and studies its leading eigenstructure. The authors prove a crossover in the bulk near $W^2\sim N$: for $W^2\gg N$ the local statistics reproduce Ginibre ensemble behavior, while for $W^2\ll N$ the correlation factorizes, indicating a localization-like regime. These results constitute a first step toward bulk universality and Anderson-type transitions for non-Hermitian random band matrices, and they establish precise spectral control via concentration near a maximum surface and Hermite-function eigenbasis analyses.

Abstract

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 $W,N\to\infty$, 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.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1
• Lemma 4.2
• Proposition 4.1
• Lemma 4.3