SUSY transfer matrix approach for the real symmetric 1d random band matrices

This paper adapts the recently developed rigorous application of supersymmetric transfer matrix approach for Hermitian 1d band matrices to case orthogonal symmetry. We consider N×N block consisting W×W random Gaussian blocks (parametrized by j,k∈Λ=[1,n]∩Z, N=nW) with a fixed entry’s variance Jjk=W−1(δj,k+βΔj,k) in each block. Considering limit W,n→∞, we prove that behaviour second correlation function characteristic polynomials such bulk spectrum exhibit crossover near threshold W∼ N.