We consider 1d random Hermitian \(N\times N\) block band matrices consisting of \(W\times W\) random Gaussian blocks (parametrized by \(j,k \in \Lambda =[1,n]\cap \mathbb {Z}\), \(N=nW\)) with a fixed entry’s variance \(J_{jk}=W^{-1}(\delta _{j,k}+\beta \Delta _{j,k})\) in each block. Considering the limit \(W, n\rightarrow \infty \), we prove that the behaviour of the second correlation function of such matrices in the bulk of the spectrum, as \(W\gg \sqrt{N}\), is determined by the Wigner–Dyson statistics. The method of the proof is based on the rigorous application of supersymmetric transfer matrix approach developed in Shcherbina and Shcherbina (J Stat Phys 172:627–664, 2018).

我们考虑由\(W\times W\) 个随机高斯块组成的1d 随机 Hermitian \(N\times N\)块带矩阵（参数化为\(j,k \in \Lambda =[1,n]\cap \ mathbb {Z}\) , \(N=nW\) ) 具有固定条目的方差\(J_{jk}=W^{-1}(\delta _{j,k}+\beta \Delta _{j ,k})\)在每个块中。考虑到极限\(W, n\rightarrow \infty \)，我们证明了这种矩阵在大部分频谱中的第二相关函数的行为，如\(W\gg \sqrt{N}\), 由 Wigner-Dyson 统计确定。证明方法基于 Shcherbina 和 Shcherbina (J Stat Phys 172:627–664, 2018) 开发的超对称传递矩阵方法的严格应用。