On the Second Mixed Moment of the Characteristic Polynomials of 1D Band Matrices

Abstract

We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of 1D Gaussian band matrices, i.e., of the Hermitian N × N matrices H N with independent Gaussian entries such that 〈H ij H lk 〉 = δ ik δ jl J ij , where $${J=(-W^2\triangle+1)^{-1}}$$ . Assuming that $${W^2=N^{1+\theta}}$$ , $${0 < \theta \leq 1}$$ , we show that the moment’s asymptotic behavior (as $${N\to\infty}$$ ) in the bulk of the spectrum coincides with that for the Gaussian Unitary Ensemble.