Delocalization of random band matrices in two and three dimensions
In plain words
Random band matrices are simplified models of disordered conductors in which each site couples randomly to all sites within a distance W. The transition between trapped and spreading states in one dimension was settled in 2025, but in 2D and 3D at fixed band width it is open.
Precise statement
Hermitian random band matrices on the box $\{1..L\}^{d}$ with variance profile of width $W$. Prove for $d = 3$ that bulk eigenvectors are delocalized for $W \ge W_0$ fixed and $L \to \infty$, and for $d = 2$ that the localization length scales as $\operatorname{exp}(c W^2)$. Answer: proofs of these scalings.
What would settle it
Proofs of delocalization for fixed large $W$ in $d = 3$ and of the $\operatorname{exp}(c W^{2})$ localization length in $d = 2$.
Status in the literature
Unverified note
In $1\mathrm{D}$ the threshold $W \sim L^{1/2}$ was established in 2025 (Yau and Yin, arXiv 2501.01718; Erdos and Riabov, arXiv 2506.06441; localization by Drogin, arXiv 2508.05802); in $2\mathrm{D}$ delocalization is proven for $W \ge L^{c}$, any $c > 0$ (Dubova, Yang, Yau and Yin, arXiv 2503.07606); fixed $W$ in $d = 2, 3$ remains open.