{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"98b1cb1351fce192edb6001d2e6dd31b37b8cff48d9b0396db168ebd044b0538","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["b8e12ec2ea1d269e7bb90c049f9bd056eb8851930b854ad7ee865b44ecea5f1d","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"f81310855cabc26fdf40b07272e276175639c6305be2847984897eeb044b3d27","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.anderson-delocalization.quantum-diffusion","kind":"well-posed","literature_status":"open","n":"2","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 $E \\sum_{x} \\mid x \\mid^{2} \\mid \\langle x \\mid \\mathrm{exp}(-itH) \\mid 0\\rangle \\mid^{2} \\sim D(\\lambda) t$ 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":"b8e12ec2ea1d269e7bb90c049f9bd056eb8851930b854ad7ee865b44ecea5f1d","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":null},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"c6ccdbe0482f9498848fe4b51fa69af971da6a01fcb21e9bab0c8db6eef4cb54","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"da4c896f832ff7e898a396b101091afd30c0840fd9e6c529a030243e4a9d2bc2","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"eC2Va8JQaYRFOrYkzZ8GtAk_WOm9dpWu95W2S0dCNyiwi4XW_FE1ROdIRmHF0PvhuNh8tUr2a-_Z_gl7x0xzAw"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEI2kyJb4Mv9-iYo5axAQka_TDAhA_Z5sUpoDAkPkqdK8LxINokig9xqOHA9Up7rrMY_yWRyepwcEbCDT6N27uvZjojCPEgeGQQLIlKJsDLLc1jmW63LODbAvtwuGOiBMs5jHB3gVII8CCEHX3ZQQtkQZG76Dtr4hqoZDxo5g40MT7TVJmuFDBIegjwIIaQXWOgbk1WxY_s7Cy5Zyml7znQxQ5m-HN2SNbwCxFRCPAg9KOsR5itmWo6z0jFQlhmpfYOxETluNUZMgC_tfmrncAI8SAkw4-FKikdZ9yrVszVracDV3BTJTboMzt2UveSyZlRtwjwIMtN_WSs-pavrta_9vKxLs54fFvTCFo_EbbiRpbJsL8_CPAg286fbogvT8QZD8nJMDWZS1r_Vyg5Jg4hl47VhsL1_UQI8SCRZP0ctidqGb7ZjgNGhZ1hJpAMc97VLUvI8DuO4xChLwjxID89HsXrUag9eRv04qSQX18D2ak5pxDN4H8m3_0Qkp84CPAg3mczMYDKOd7Nhf3ojA9ZvuTsN66Y1NnyYTp9j8GKQNwI8SCvAiiN-Hrv7IEejd8wdUihoozu5KOvpoBFJ0vg3UnGAQj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIGmWUfjFVlOvsLDpNDdIhWVJKi8Pt3qdnx1jmCNi_G5rxIBpC7CsLZKO0QlAzpv2JT8OuxKtNq6YwAJA1KBE6HxkACPAg-Dd1MzVKLWRVbACv4B9VMTcNijQezjw2mw0uD9IpwQ0I8CClKdr7B-7Ok-QQ6D2FGpowADs_gX12CvAptPjOkir7agjxIK4S8yeen2Z9ad51dB-w9vA_MkkQKoZ9nneKN_JEDPm_CPAgILN14VTArZz6q9uCt8e3t6CncR3OEgjx9JnysCmr-ZMI8SB5cYmCf4U2vqCDk8Vz2kdoUwnLtzufB7a_lu69qsYQoAjxILNBbiMXjFx2nLAsqq2yLOEYPPpEYUfXyZmHy3ijHVMeCPAgQEjWwWlWWX3vJoLxfq0lixNb2cwGMRGhB0ekxjFJDZAI8SCHI26f1bMr-H5VtXG_HgMHDd8PPImPLhsWPB8PPaY3jwjwIKD8mGRUgbk2Dt1xG-4OxMpCuw1ya5UEUQXsqASURGCtCPAgkPXX-naXR0fRIdS-sEDlzYmtMvSCNH42ESUF0QXb6NUI8CDQo_sIWZiWXaTGRSQBLmU-lHJKskYKqsjO1r02DhB8LAj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"]},"log":{"leaf_index":"3400","tree_size":"3804","proof":["37fa8b7c01657553fb39c991e5769ff459a95b0d7366de784dab626b3c7c207b","88f9e44ca27e17a102f8c87033817e0b9d80ae57b575e1ac0d9ecd48ed06fad7","f881bf90427f006156ace03b368160825626a64458e1d151ef17b6585ec2fe83","64233a076453971cc2b94fc426c1013b594a0856b8082ba4ba4b3b87dd565784","f8054e07d574f827bbe7eaba51dfcf5e57fc972d58b070fa0a72f3c754638d24","9d431ce7742af24a57989023456a175d95e430366c461878574a00aa046e43b8","217966f7b365dc70af6d616a83ae450e616d3fef50ead5372fb26577cf221836","f543c2acee32c00eed64ca6a002c4cdf20cd84f300a60f431863f4556659c7bc","5bc8f1bbb7a36ffa2afc301ec767424419006f77d79dfc7ac3adb0dab9eb36e8","98b5a14b42b3a202b6ef825f66833fa505601404c4c0f14575f3ec922607a858","af34d80d0065c0ab502282775e865492e2cb49c9493f4594d63ba5577922854a","3c578dccc15502c5f3efe228cadc99fee76ab536ed01f66b14d316fdcc157429"],"checkpoint":"pubphys.com/log/v1\n3804\nAK6t4qgu96+M9wScTrIlsJ10d/zhKVEtYguyM3aeJvo=\n\n— pubphys.com/log/v1 JFPwIPP8EtoxgcsQg7V7CwIdL4UKPqJtpv+eB8sFNJjyQHIdyA1VabmBsPyLqq2bLxJPn57rA6fQi8y0QDDjRXHvlQg=\n","promise":"pubphys.com/log/v1 promise\n0bb5c4ceb5012c635b7ef0d2e6f8327b541b9730bde6f9744359bd78e12d5b2e\n3400\n2026-10-15T23:43:57Z\n2026-10-16T01:43:57Z\n\n— pubphys.com/log/v1 JFPwIJG1MWtomx65k9+nq1o8Yk6Zn9Ztbg8QMzuxEhPsjJKcOUceFwVPJyWwkhWNif4F9L0zupo+Az3Ww67rp/PNewY=\n"},"orcid_key_evidence":[],"attesting":null}