{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"b5c37212240798b5323e2144d0fb218573193b98651c6222fe7143d72f16e581","created":"2026-10-03T07:18:11Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"e7373c12ad32cc6a17d4d74f1dcaec82aa5c14c69369ae451543417c6612c51e","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"stat.kpz-higher-d.upper-critical-dimension","kind":"well-posed","literature_status":"contested","n":"1","parents":[],"plain":"In high enough dimension many fluctuating systems become simple and are described exactly by mean-field theory. It is unknown whether KPZ growth has such a dimension, above which the rough phase loses its nontrivial roughness, or whether nontrivial roughness persists in every dimension.","posed_since":"","precise":"Determine whether there is a finite $d_u$ such that for $d \\ge d_u$ the strong-coupling KPZ fixed point has $\\alpha = 0$ and $z = 2$ (mode-coupling and some functional RG treatments suggest $d_u = 4$), or whether $\\alpha > 0$ for every finite $d$. Answer: yes with $d_u$, or no.","problem_ref":null,"references":"","settled_by":"A controlled theory of the strong-coupling fixed point in general d, confirmed by simulations of growth models or directed polymers in $d = 4\\text{ to }6$ that resolve whether $\\alpha$ vanishes.","status_note":"Simulations up to about $d = 5$ find nonzero $\\alpha$, against theories with $d_{u} = 4$; a 2026 study examines the infinite-dimensional limit on fully connected graphs (arXiv:2603.02000).","title":"Upper critical dimension of the KPZ 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