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

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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.

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Correspondence to Tatyana Shcherbina.

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Communicated by L. Erdös

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Shcherbina, T. On the Second Mixed Moment of the Characteristic Polynomials of 1D Band Matrices. Commun. Math. Phys. 328, 45–82 (2014). https://doi.org/10.1007/s00220-014-1947-7

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