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 $W\to \infty$ 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 $\beta ,n\to \infty$, 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.

38 pp

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