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The paper continues (Shcherbina and Shcherbina in Commun Math Phys 351:1009–1044, 2017); Shcherbina in Commun Math Phys 328:45–82, 2014) which study the behaviour of second correlation function of characteristic polynomials of the special case of n × n one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix J = (- W 2 ▵ + 1) - 1 . Applying the transfer matrix approach, we study the case when the bandwidth W is proportional to the threshold n . [ABSTRACT FROM AUTHOR]

We prove that the asymptotic behavior of the second mixed moment of the characteristic polynomials of the N × N 1D Gaussian real symmetric band matrices with the width of the band W ≫ N1/2 coincides with those for the Gaussian orthogonal ensemble. Here, we adapt the approach of Shcherbina [Commun. Math. Phys. 328, 45.82 (2014)], where the Hermitian case was considered. [ABSTRACT FROM AUTHOR]

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 with independent Gaussian entries such that 〈 H H〉 = δδ J, 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. [ABSTRACT FROM AUTHOR]

We consider the asymptotic of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices H = n W. We show that for the correlation function of any even order the asymptotic coincides with this for the Gaussian Unitary Ensemble up to a factor, depending only on the fourth moment of the common probability law of entries $${\mathfrak{J} W_{jk}, \mathfrak{R} W_{jk}}$$ , i.e. that the higher moments do not contribute to the above limit. [ABSTRACT FROM AUTHOR]