{"schema":"pubphys.bundle/1","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 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