Universality for 1d Random Band Matrices: Sigma-Model Approximation

The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:1233–1260, 2016; Commun Math Phys 351:1009–1044, 2017). We consider random Hermitian block band matrices consisting of W× W random Gaussian blocks (parametrized by j, k∈ Λ = [1 , n] ^d∩ Z^d) with a fixed entry’s variance J_jk= δ_j,kW^{- 1}+ βΔ _j,kW^{- 2}, β> 0 in each block. Taking the limit W→ ∞ with fixed n and β, we derive the sigma-model approximation of the second correlation function similar to Efetov’s one. Then, considering the limit β, n→ ∞, we prove that in the dimension d= 1 the behaviour of the sigma-model approximation in the bulk of the spectrum, as β≫ n, is determined by the classical Wigner–Dyson statistics.