{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"a62ead620a3df6a2d469081fbe931f3957881d52f5f768c3dc468960dceebb42","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["072646902f99b00518a78e01f67a64caa37955b3543a6ca997ef80e374b0ece4","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"2e99eafe8abd1436c6ad2966da0bbe545c0f0f09a1cffacafcf464e484285b6f","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":"2","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 $\\langle s_{0} . s_{x}\\rangle \\le C\\ \\exp(-m(\\beta)\\ \\mid x\\mid)$ 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":"072646902f99b00518a78e01f67a64caa37955b3543a6ca997ef80e374b0ece4","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":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"217d7022c4740e68ebf95399c8a720e31664f4d542e31ac924ae6f46a677be09","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"1bfa47259d64e289acce9c0c7a3f6e890c89afc440ab1b48490ac0fba89b49de","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"mwpuZvDAmkt5rviFJv47mpZxLdZB_1N6dAdo7ADTRy5vD0PcMZRtmbMI_75knS_S0VICshiR9ZYlnUvWrNG1Ag"},"schema":"pubphys.envelope/1"},"record_hash":"217d7022c4740e68ebf95399c8a720e31664f4d542e31ac924ae6f46a677be09","leaf_index":3454}