{"schema":"pubphys.bundle/1","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},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"217d7022c4740e68ebf95399c8a720e31664f4d542e31ac924ae6f46a677be09","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"1bfa47259d64e289acce9c0c7a3f6e890c89afc440ab1b48490ac0fba89b49de","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"mwpuZvDAmkt5rviFJv47mpZxLdZB_1N6dAdo7ADTRy5vD0PcMZRtmbMI_75knS_S0VICshiR9ZYlnUvWrNG1Ag"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIG_pHJZ1k4omszpwMej9uiQyJr8RAqxtISQrA-6ibSd7xIBvzHfRgngXt5UoQc1-CyANG6DuD4lcU_QlrhdVXoluBCPEgOD4FzKbfK0mKezDf72L5IKxmwUAXiBamhyRDKYKdavMI8SBzAdWi0tH_f7GsvFA_RuPhdoozX_8JWnXMEh4rWixivwjxINrBFT2IS1nsI-NOVHpsIBep7zt3vd89jk7wNr-Lytl6CPEgJpG-5wfY2-QlkahmwmckBcXNn_684e3l3NJQErqRmNwI8SB5cYmCf4U2vqCDk8Vz2kdoUwnLtzufB7a_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"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEI18m-TaBJ02uvJTM7Dndzjn89oN1HekhKMZuldZFcIZvwINfgF77_j_0NBiFBcyUteMx-PCBTVC7PXl87Ciu0sWS_CPEgFRus8VKNK8kq0073s477SB2jsceakPJkFU4agU53ER0I8CD5ecocl99RmVukPHvZyKgDnNT5KXZWSRNjlykGSkGGYgjwIFx0G2uxQVTpO_oba7wZ26suJz_okAQ3TibSqQrInpLLCPAgVd5C38oSm90KLhwkqeh-uc1uMVvB7cwHFeoszZDC62oI8CBUXj9yD2WgHsKs4RFyQf0pkA70wIaTjGlBxyZ2FK0q8AjwIMtN_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"]},"log":{"leaf_index":"3454","tree_size":"3804","proof":["fc791921e091fbc2c9b98b4ef11e72eb24f47d14608591c9f66d567e441c36be","b1208de5969102522b0bfcd60e0c9ee151081abbc976b61aec7830f2f1f9c985","77a764e0773f6aa5ac287346bbfa0ca7fa2cf1e226a0b578f61fdae96a88fcad","c1c4a3513036c99b4ee08de59e90fb3af34c67e9fa64c79871d38838af5e0345","b97e5f8ea13e637caddc43bf95c4486664ff6f2e9a6fc9f825a26eece8c42a49","13aebc7976993ff4cffdacbb303db40c3f168539f7c2261b2aa55d2ad5dfe5e0","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\n68779b3853665f77f73b82019e4d433f04b82d4e11d40cb34a4978251614f89f\n3454\n2026-10-15T23:43:57Z\n2026-10-16T01:43:57Z\n\n— pubphys.com/log/v1 JFPwIE6ksDpyKFpu5vMHC2glRiigLqm5s58e6EsWvsl4JSftiHoqu3YBiJ5C78PNmlSMXx05NzKdXaeYvwmG81rE9wY=\n"},"orcid_key_evidence":[],"attesting":null}