{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"0d8c86813ab66522c4e41848d77cfa29aad24fe12f1c12fd33f46485b7e02ad2","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"cd937a5c0f8573272070ba4a7644aa766758ec6ed2e4239d78d143d7e5ec37ef","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"mathph.sigma-models.polyakov-mass-gap","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Prove that in a flat grid of three-component (or more) arrows, the correlation between two arrows dies off exponentially with their distance at every temperature above zero. For two-component arrows this is false at low temperature, which is why the claim is subtle.","posed_since":"1975","precise":"Classical $O(N)$ model on $Z^{2}$, $N \\ge 3$, spins $s_{x}$ in $S^{N-1}$, Gibbs weight exp(beta sum_<xy> s_x . s_y). Prove that for every $\\beta < \\infty$ there are $C, m(\\beta) > 0$ with $\\langle s_{0} . s_{x}\\rangle \\le C\\ \\operatorname{exp}(-m(\\beta)\\ \\mid x\\mid)$ uniformly in the volume. For $N = 2$ the opposite holds at large $\\beta$ (power-law decay, Frohlich and Spencer, 1981). Answer: a proof, or a proof of a massless low-temperature phase.","problem_ref":null,"references":"","settled_by":"A proof of a positive mass $m(\\beta)$ for all finite $\\beta$, or a proof of power-law decay at some finite $\\beta$, for $N = 3$.","status_note":"Exponential decay is proven only at small $\\beta$ by high-temperature expansion; Patrascioiu and Seiler argued for a massless low-temperature phase, and percolation properties of the $2D$ Heisenberg model were studied by Aru, Garban and Sepulveda (arXiv 2212.06767, 2022).","title":"Exponential decay of correlations in the 2D O(N) model at all temperatures","topic_ref":"b4bb48d722ff2368d355af570ae350e795f3cc5cea7a0142509f239da3c98279"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"072646902f99b00518a78e01f67a64caa37955b3543a6ca997ef80e374b0ece4","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"fd6a9c44cf1b0700406c2a2111317ecce5e0d0ad3067659019e3fffbf2000a82","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"HMIfnvrA1PPy8E7iITVFY1Tg5BLIXWWeDkmb9t8FWcozPMj04nTJlwVzHlOjV4iK3kd5J4a_bbI7eJT7G1vICQ"},"schema":"pubphys.envelope/1"},"record_hash":"072646902f99b00518a78e01f67a64caa37955b3543a6ca997ef80e374b0ece4","leaf_index":1658}