Characteristic polynomials for 1D random band matrices from the localization side

TL;DR

This paper investigates the behavior of characteristic polynomials in 1D Gaussian Hermitian random band matrices, revealing a crossover phenomenon at a critical band width, which differs from classical GUE results.

Contribution

It proves the limiting behavior of the second mixed moment of characteristic polynomials for 1D random band matrices when the band width is much smaller than the matrix size, demonstrating a crossover at W ~ sqrt(n).

Findings

Limit of normalized second mixed moment approaches one for W << sqrt(n).

Results differ from GUE, indicating a crossover at W ~ sqrt(n).

Supports the localization theory in 1D random band matrices.

Abstract

We study the special case of $n\times n$ 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by $J=(-W^2\triangle+1)^{-1}$. Assuming that the band width $W\ll \sqrt{n}$, we prove that the limit of the normalized second mixed moment of characteristic polynomials (as $W, n\to \infty$) is equal to one, and so it does not coincides with those for GUE. This complements the previous result of T. Shcherbina and proves the expected crossover for 1D Hermitian random band matrices at $W\sim \sqrt{n}$ on the level of characteristic polynomials.