Localization Lengths for Two-Dimensional Random Band Matrices: Stretched-Exponential Lower Bounds
Year: 2026
Type: preprint
Abstract: We consider $N \times N$ random band matrices $H = (H_{xy})$ with centered complex Gaussian entries, indexed by points $x,y$ on the two-dimensional discrete torus $(\mathbb{Z} / \sqrt{N} \mathbb{Z})^2$. The matrix entries $H_{xy}$ vanish whenever the distance between $x$ and $y$ exceeds the bandwidth parameter $W$. We prove that if $W \geq (\log N)^{7}$, then, with high probability, all bulk eigenvectors are delocalized. Equivalently, this yields a lower bound of order $\exp(W^{1/7})$ for the localization lengths of two-dimensional random band matrices, improving the superpolynomial lower bound $W^C$ for every fixed constant $C>0$ established in arXiv:2503.07606.
Source: arXiv (Cornell University)
Authors: Lai, Xujie, Yang, Fan
Institutions: Institute of Mathematical Sciences, Tsinghua University
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214031335
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