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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.","posed_since":"","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 $\\exp(c W^2)$. Answer: proofs of these scalings.","problem_ref":"6ec1685f80851ecf07cfb91f964af6cf574af34e443f78b78d0a943108f88fab","references":"","settled_by":"Proofs of delocalization for fixed large $W$ in $d = 3$ and of the $\\exp(c W^{2})$ localization length in $d = 2$.","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.","title":"Delocalization of random band matrices in two and three 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