{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"107497d217e829ad5f16b3a770ef9cfc11106517033f4b4530981fb10b965add","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["6ec1685f80851ecf07cfb91f964af6cf574af34e443f78b78d0a943108f88fab","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"59d0542c7493ff83de10ea0667e151180cfa28073994136daac23b4d4914bfc2","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":"2","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 $\\exp(c W^2)$. 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 dimensions","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"3e03127bc8f8cf419e66f1ea76d023a722fd573fa7ed16044b054f900981dd9f","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"d85923d5850fab5a0d46b08533d1402e242ee2d4e0c7d1d5b6467096d81ba84c","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"8dlQr3XP7B8JzK7M010BxOggFCYJ8iq3JyX6R9rWRt7voZ8zu_23vBpowcpefUy7DRZKFNK8b0vwH8fhhAj_AA"},"schema":"pubphys.envelope/1"},"record_hash":"3e03127bc8f8cf419e66f1ea76d023a722fd573fa7ed16044b054f900981dd9f","leaf_index":3398}