{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"1a51a10c00976c24977670c5059c88f0afc496a0e27e8c8c6fb3f4121838fd65","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"6aa7afcb4f90053e76ac34291bf957056d46a73d75634ab9e99d78051d38db06","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.anderson-delocalization.quantum-diffusion","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"An electron in a weakly disordered 3D crystal should wander like a random walker, with its mean squared distance growing linearly in time forever. This is proven only up to a long but finite time.","posed_since":"","precise":"For $H = \\Delta + \\lambda V$ on $Z^3$, $\\Delta$ the nearest-neighbour hopping operator, show that E sum_x |x|^2 |<x|exp(-itH)|0>|^2 ~ D(lambda) t as $t \\to \\infty$ with a diffusion constant $D(\\lambda) > 0$ for small $\\lambda$ (or the analogous statement for an energy-localized initial state). Erdos, Salmhofer and Yau proved diffusion up to times $t \\sim \\lambda^{-2-\\kappa}$ for some $\\kappa > 0$. Answer: a proof for $t \\to \\infty$ at fixed small $\\lambda$.","problem_ref":null,"references":"","settled_by":"A proof of diffusive spreading uniformly in time at fixed small $\\lambda$.","status_note":"Diffusion is controlled only up to times $\\lambda^{-2-\\kappa}$ (Erdos, Salmhofer and Yau, around 2007-2008).","title":"Quantum diffusion for all times in the weakly disordered 3D lattice","topic_ref":"401a78c75cbf7b8acc9217ce7c5ab1fcd800a8c8cb9b4f7b06d2e6e76f0bedcb"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"b8e12ec2ea1d269e7bb90c049f9bd056eb8851930b854ad7ee865b44ecea5f1d","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"4f0513b14cd09787626b005b6ee03c88fbcea8f07abd6c500af668b78812eede","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"1blHOGW3tOX8ti4MIN93uTqkTeI54UVJ33Xa7aD0OqoX-g3oc2o-RPeXCv0qI_7Dajd4dGbCIHe05jLP8BycBQ"},"schema":"pubphys.envelope/1"},"record_hash":"b8e12ec2ea1d269e7bb90c049f9bd056eb8851930b854ad7ee865b44ecea5f1d","leaf_index":1595}