{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"9174ba39a4be52fe16e64c4bf27037bf594499f62558fe5a378d856d468fb23f","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["a2ff45b3870647d7740f34e6c59fcaa8d4dca1ef67d9ec99d0add5cbd9796987","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"da4e8cb68e1387c8a7b3f784a23952a99fb876f4b5dafebe45262720e942ca94","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.anderson-delocalization.2d-localization","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"Scaling theory predicts that in two dimensions every electron state is trapped by any amount of disorder, however weak. A proof exists only near the edges of the energy band or at strong disorder.","posed_since":"1979","precise":"For $H = \\Delta + \\lambda V$ on $\\ell^2(Z^2)$, $\\Delta$ the nearest-neighbour hopping operator (spectrum $[-4, 4]$ at $\\lambda = 0$), $V$ i.i.d. with bounded density (or Bernoulli), prove that for every $\\lambda > 0$ the spectrum is almost surely pure point with exponentially decaying eigenfunctions at all energies, with localization length expected to grow as $\\exp(c/\\lambda^2)$. Answer: a proof.","problem_ref":"a2ff45b3870647d7740f34e6c59fcaa8d4dca1ef67d9ec99d0add5cbd9796987","references":"","settled_by":"A proof of complete localization for small $\\lambda$ including the band center.","status_note":"Localization is known at large disorder and near band edges, including for Bernoulli disorder near the edge (Ding and Smart, 2020).","title":"Localization at all energies in the weakly disordered 2D Anderson model","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"dc012abc4937aa9789e4fa5ecf88a43f75acc284c0ef7cb4c282d77c9f18fffc","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"b3c1a0153f5e9e8fddd734aceae3ee714acb9e4c7af7ba53779da386fdbf55a4","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"cxJkfBOYd3imNGdKjDoz3_IOKg_y1fBZ6fsBasLJGQX8TjKyi06PyHqezb63FNJpUv_PuAwqWrQxROowttgTAw"},"schema":"pubphys.envelope/1"},"record_hash":"dc012abc4937aa9789e4fa5ecf88a43f75acc284c0ef7cb4c282d77c9f18fffc","leaf_index":3396}