SUSY 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 Hermitian 1d band matrices to the case of the orthogonal symmetry. We consider N × N block band matrices consisting of 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 + β∆j,k) in each block. Considering the limit W, n → ∞, we prove that the behaviour of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold W ∼√N.