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arXiv · 2011.13457

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

Tatyana Shcherbina
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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 $W\sim \sqrt{N}$.

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Tatyana Shcherbina. 2020-11-26. Transfer matrix approach for the real symmetric 1D random band matrices. https://arxiv.org/abs/2011.13457

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