{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"0317007e5c30c65737e41b163602e2fd29876a0818f883c20f16ec1b8e787008","created":"2026-10-03T07:18:11Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"7c7d673d580b7328740b4371095f2c01306bf53556d4c86f80507fcb3cccd2e4","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"stat.synchronization-networks.lattice-lower-critical-dimension","kind":"well-posed","literature_status":"contested","n":"1","parents":[],"plain":"When each oscillator is coupled only to its neighbors on a lattice and natural frequencies are random, synchronization may be impossible in low dimensions. The dimensions above which frequency locking and full phase locking can occur are not established.","posed_since":"1987","precise":"For Kuramoto oscillators on a $d$-dimensional hypercubic lattice with nearest-neighbor coupling $K$ and i.i.d. natural frequencies of finite variance, determine the lower critical dimension $d_f$ for macroscopic frequency entrainment and $d_p$ for phase synchronization (nonzero global order parameter) at finite $K$ as $N \\to \\infty$. Fluctuation arguments suggest $d_f = 2 \\text{ and } d_p = 4$; numerical evidence is mixed. An answer is the pair $(d_f, d_p)$ with proof or controlled numerics.","problem_ref":null,"references":"","settled_by":"A proof of the absence or presence of entrainment and phase order in each $d$, or simulations with finite-size scaling in $d = 2\\ \\text{to}\\ 5$ that fix both dimensions.","status_note":"","title":"Lower critical dimensions for synchronization on lattices","topic_ref":"e69da0d5ffbbc9da951a478d590c05a19d606b183df93f87687dfb38b6e26853"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"2e4535f6a571b869a601f10a298e09b82916f0a1d0c974bd0328541f519299cf","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"6fdaef456c5055eb2b7f23c07ed3267f61aa16813d253bb9093f91887628a214","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"7fvUVzzcAPwA4Zlqfqjt5HJ_Yb6gKrB9ELeNGUGxxKIJLeWRLhIV25qARSp2BbQ_rDCSbjys7BmIHRBZHq79Ag"},"schema":"pubphys.envelope/1"},"record_hash":"2e4535f6a571b869a601f10a298e09b82916f0a1d0c974bd0328541f519299cf","leaf_index":2121}