Localization Lengths for Two-Dimensional Random Band Matrices: Stretched-Exponential Lower Bounds

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 $(\mathbb{Z}\mathrm{/}\sqrt{N}\mathbb{Z}{)}^2$. The matrix entries $H_{xy}$ vanish whenever the distance between $x$ and $y$ exceeds the bandwidth parameter $W$. We prove that if $W\ge (\log N{)}^{55}$, then, with high probability, all bulk eigenvectors are delocalized. Equivalently, this yields a lower bound of order $\exp (W^{1\mathrm{/}55})$ 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.