Transfer matrix approach for the real symmetric 1D random band matrices

TL;DR

This paper extends the transfer matrix approach to orthogonal symmetry for 1D random band matrices, analyzing the spectral correlation behavior and identifying a crossover near a specific matrix size threshold.

Contribution

It adapts the supersymmetric transfer matrix method to orthogonal symmetry and studies spectral correlations in large 1D band matrices.

Findings

Spectral correlation functions show a crossover near W ~ sqrt(N).

Behavior of characteristic polynomial correlations is characterized in the bulk spectrum.

The approach confirms the universality class transition in spectral statistics.

Abstract

This paper adapts the recently developed rigorous application of the supersymmetric transfer matrix approach for the 1d band matrices to the case of the orthogonal symmetry. We consider $N\times N$ block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k \in\Lambda=[1,n]\cap \mathbb{Z}$, $N=nW$) with a fixed entry's variance $J_{jk}=W^{-1}(\delta_{j,k}+\beta\Delta_{j,k})$ in each block. Considering the limit $W, n\to\infty$, we prove that the behavior of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold $W\sim \sqrt{N}$.