{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"829578460eee45256d78274025c01c0ad7e9bf0651e9742dfacb3a3dc7965db1","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"9e1e033dd4c9d8033c24d912ee7bd28bed0486f9b28a48ee174780dc2b32faea","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.anderson-delocalization.band-matrices","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"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.","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 $\\operatorname{exp}(c W^2)$. Answer: proofs of these scalings.","problem_ref":null,"references":"","settled_by":"Proofs of delocalization for fixed large $W$ in $d = 3$ and of the $\\operatorname{exp}(c W^{2})$ localization length in $d = 2$.","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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