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Title: Delocalization of random band matrices in two and three dimensions
Plain: 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: 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.
Settled by: Proofs of delocalization for fixed large $W$ in $d = 3$ and of the $\operatorname{exp}(c W^{2})$ localization length in $d = 2$.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: partially-resolved
Status 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.
Posed since: (empty)