{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"dee91799af4591f78b019f842a2836d22c67ba88d988b426cebbd7fc227b331b","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"9565e199c8efe7917e2c469f3d2f5b8bc9d672e65cf6d652b84dca33e1c6d9e0","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":"1","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 $\\operatorname{exp}(c/\\lambda^2)$. Answer: a proof.","problem_ref":null,"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":"401a78c75cbf7b8acc9217ce7c5ab1fcd800a8c8cb9b4f7b06d2e6e76f0bedcb"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"a2ff45b3870647d7740f34e6c59fcaa8d4dca1ef67d9ec99d0add5cbd9796987","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"7c4c075a888499567e2b6bf86d6ed2d2cc2bdfe5705685b80335d3ced03960d9","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"u5DxLPbsQFnDUT4yPjGzG5VdnpZuLxLd2mXPED1hlVY8YOGy8Jg4O2YsXxO6bXoS4bvjLmDoLnup_AK37nuaBg"},"schema":"pubphys.envelope/1"},"record_hash":"a2ff45b3870647d7740f34e6c59fcaa8d4dca1ef67d9ec99d0add5cbd9796987","leaf_index":1591}