Strong KPZ universality conjecture

Establish that, for a large class of random interface growth processes, the height functions under the 3:2:1 time-space-fluctuation scaling converge to a universal random field known as the KPZ fixed point.

Background

The paper places its one-dimensional random growth surface within the broader KPZ universality class. It distinguishes the subcritical Edwards–Wilkinson limit studied in the paper from the stronger KPZ fluctuation regime, in which nonlinear effects persist and the characteristic 3:2:1 scaling is expected to produce the KPZ fixed point.

The stated conjecture concerns universality across a large class of random interface growth processes, rather than only the discrete recursion analyzed in this paper. The paper cites the KPZ fixed-point literature as the context for this unresolved universality claim but does not attempt to prove it.

References

A central challenge is to identify the counterpart of a dilute population edge on such networks and determine whether its fluctuations admit an effective KPZ description.

— Anomalous First Passage in Evolution: Edge-KPZ Theory  (2609.16499 - Hatakeyama, 15 Sep 2026) in Section Discussion, paragraph beginning “Our theory could help explain how rapidly populations gain access to new phenotypes”

A central goal is to understand the strong KPZ universality conjecture, which states that for a large class of random interface growth processes, the height functions under the $3:2:1$ time-space-fluctuation scaling should converge to a universal random field, also known as the KPZ fixed point .

— Subcritical limits of a 1D KPZ growth surface with finite moments  (2609.08308 - Liu et al., 8 Sep 2026) in Section 1, subsection “Background and related results”