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

Abstract

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 $$W\times W$$ random Gaussian blocks (parametrized by $$j,k \in \Lambda =[1,n]^d\cap \mathbb {Z}^d$$ ) with a fixed entry’s variance $$J_{jk}=\delta _{j,k}W^{-1}+\beta \Delta _{j,k}W^{-2}$$ , $$\beta >0$$ in each block. Taking the limit $$W\rightarrow \infty $$ with fixed n and $$\beta $$ , we derive the sigma-model approximation of the second correlation function similar to Efetov’s one. Then, considering the limit $$\beta , n\rightarrow \infty $$ , we prove that in the dimension $$d=1$$ the behaviour of the sigma-model approximation in the bulk of the spectrum, as $$\beta \gg n$$ , is determined by the classical Wigner–Dyson statistics.