We consider the deformed Gaussian ensemble H n = H (0) n +M n in which H (0) n is a hermitian matrix (possibly random) and M n is the Gaussian unitary random matrix (GUE) independent of H (0) n . Assuming that the Normalized Counting Measure ofn converges weakly (in probability if random) to a non-random measure N (0) with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of H n .

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 △ + 1) −1 . Assuming that W 2 = N 1+θ , 0 < θ ≤ 1, we show that the moment's asymptotic behavior (as N → ∞) in the bulk of the spectrum coincides with that for the Gaussian Unitary Ensemble.

We study the special case of n × n 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by J = (−W 2 △ + 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 coincides with those for GUE. This complements the result of [18] and proves the expected crossover for 1D Hermitian random band matrices at W ∼ √ n on the level of characteristic polynomials.

We consider the block band matrices, i.e. the Hermitian matrices H N , N = |Λ|W with elements H jk,αβ , where j, k ∈ Λ = [1, m] d ∩ Z d (they parameterize the lattice sites) and α, β = 1, . . . , W (they parameterize the orbitals on each site). The entries H jk,αβ are random Gaussian variables with mean zero such that. This matrices are the special case of Wegner's W -orbital models. Assuming that the number of sites |Λ| is finite, we prove universality of the local eigenvalue statistics of H N for the energies |λ 0 | < √ 2.