Universality for 1d random band matrices: sigma-model approximation

TL;DR

This paper extends the supersymmetric transfer matrix approach to 1D random band matrices, deriving a sigma-model approximation and showing that in certain limits, the spectral statistics follow classical Wigner-Dyson behavior.

Contribution

It develops a rigorous sigma-model approximation for 1D random band matrices and proves universality of spectral statistics in the large limit.

Findings

Sigma-model approximation derived for 1D random band matrices.

In the limit, spectral statistics match Wigner-Dyson distribution.

Results confirm universality in the bulk spectrum for large parameters.

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.