{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"3c2337e1e551d2a0a079f291cec8bf07f80eca003f7212b2d5385cb42933ef5a","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"fa3dbbf69bbfbe8fd2b31f231bd4417813c9e8d96abb3e0bf80439dd6f285d43","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.ising-rigorous.saw-3d","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"A random path that never visits the same point twice models a polymer chain, and in 3D its size should grow as the number of steps to the power 0.588. Even a power-law improvement over straight-line growth is unproven.","posed_since":"","precise":"Self-avoiding walk on $Z^3$ with $n$ steps, uniform measure. Prove $\\langle \\mid w(n)\\mid^2\\rangle^{1/2} = n^{\\nu+o(1)}$ with $\\nu\\ \\text{approximately}\\ 0.5876$, or at least $\\langle \\mid w(n)\\mid^2\\rangle^{1/2} \\le n^{1-\\delta}$ for some $\\delta > 0$. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A polynomial sub-ballistic bound, then existence of $\\nu$.","status_note":"Sub-ballistic behavior $o(n)$ is proven (Duminil-Copin and Hammond, 2013); in $d = 4$ weakly self-avoiding walk has proven logarithmic corrections (Bauerschmidt, Brydges and Slade).","title":"Endpoint-distance exponent of the 3D self-avoiding walk","topic_ref":"0b88a0ded9bd47630f1a876bef355f9b17efdebced19bd55e73bb1cd79f5cae3"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"05c5862b9e4c60a6c12d015a3fd71f7a1e7b5dfdea72d8b85299db1b22c69ca5","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"261dcfccbba76f4d1346e70c36f44c875db5c02bca205dd67e48617def649844","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"KLFU8rhEevXjbQXAn3uzb1KIMZeGD17Lgss9fPERPSphk_LB9J6nAglmDjhSDPVBF8WINRN5jdcITOordCHSDA"},"schema":"pubphys.envelope/1"},"record_hash":"05c5862b9e4c60a6c12d015a3fd71f7a1e7b5dfdea72d8b85299db1b22c69ca5","leaf_index":1632}