Upper critical dimension of the KPZ class
In plain words
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.
Precise statement
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.
What would settle it
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 in the literature
Unverified 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).