Scale invariance at the KPZ roughening transition in three dimensions
In plain words
Above two dimensions a KPZ surface stays smooth when the slope effect is weak and becomes rough when it is strong. A 2021 proposal says the transition point between the two lacks ordinary scale invariance and instead repeats its pattern only at discrete zoom factors.
Precise statement
For the KPZ equation in $d = 3$ at the critical coupling $g_c$ separating the smooth (Edwards-Wilkinson) phase from the rough phase, equivalently the weak-to-strong disorder transition of directed polymers in $3+1$ dimensions, determine whether the transition is a scale-invariant fixed point (perturbative RG gives $z = 2$ and $\alpha = 0$ there) or shows discrete scale invariance with log-periodic corrections, produced by a relevant three-body coupling of Efimov type (three-boson binding with a geometric spectrum) through the mapping to attractive bosons at their binding transition. An answer is a classification, with the log-periodic period if present.
What would settle it
Simulations of directed polymers or growth models in $d = 3$ at the transition precise enough to detect or exclude log-periodic oscillations in free-energy or height fluctuations, or an RG computation that includes three-body operators.
Status in the literature
Nakayama and Nishida proposed Efimov-type discrete scale invariance at the transition (Phys. Rev. E 103, 012117, 2021, arXiv:2010.15161); no numerical test of it has been located.