Weak universality conjecture for one-dimensional random growth surfaces

Prove that a class of one-dimensional random growth surfaces driven by microscopic fluctuations, with local growth depending nontrivially on neighboring heights, converges to the KPZ equation under appropriate scaling.

Background

The weak universality conjecture concerns discrete one-dimensional growth surfaces whose local nonlinear growth depends on neighboring heights. It predicts convergence to the one-dimensional KPZ equation after a suitable macroscopic rescaling.

This conjecture is broader than the subcritical result established in the paper: the paper proves convergence to the additive stochastic heat equation in a subcritical regime, where nonlinear contributions become negligible. The conjectured KPZ limit concerns the regime and scaling in which the nonlinear behavior remains relevant.

References

The weak universality conjecture is that a class of 1D random growth surface driven by microscopic fluctuations, with the local growth depending nontrivially on neighboring heights should converge to the KPZ equation under appropriate scaling.

— 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”