Characteristic Polynomials for 1D Random Band Matrices from the Localization Side

We study the special case of n× n 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by J=(-W2▵+1)-1. Assuming that the band width W≪n, we prove that the limit of the normalized second mixed moment of characteristic polynomials (as W, n→ ∞) is equal to one, and so it does not coincide with that for GUE. This complements the result of Shcherbina (J Stat Phys 155(3):466–499, 2014) and proves the existence of the expected crossover for 1D Hermitian random band matrices at W∼n on the level of characteristic polynomials.