Transfer matrix approach for the real symmetric 1D random band matrices

Abstract: This paper adapts the recently developed rigorous application of the supersymmetric transfer matrix approach for the 1d band matrices to the case of the orthogonal symmetry. We consider $N\times N$ 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 $W, n\to\infty$, we prove that the behavior of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold $W\sim \sqrt{N}$.