# Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates

## TL;DR

This paper investigates the transient and long-term interface statistics of 1D KPZ systems with substrates shrinking or expanding over time, revealing a universal transient flat regime regardless of curvature.

## Contribution

It demonstrates that large initial size systems exhibit a transient flat KPZ statistics, and generalizes the Family-Vicsek scaling for shrinking interfaces, clarifying previous experimental controversies.

## Key findings

- Transient flat KPZ statistics observed in large initial size systems
- Crossover from flat to curved KPZ statistics at long times for expanding systems
- A generalized scaling law for roughness in shrinking KPZ interfaces

## Abstract

Recent experimental works on one-dimensional (1D) circular Kardar-Parisi-Zhang (KPZ) systems whose radii decrease in time have reported controversial conclusions about the statistics of their interfaces. Motivated by this, we investigate here several 1D KPZ models on substrates whose size changes in time as $L(t)=L_0 + \omega t$, focusing on the case $\omega<0$. From extensive numerical simulations, we show that for $L_0 \gg 1$ there exists a transient regime in which the statistics is consistent with that of flat KPZ systems (the $\omega=0$ case), for both $\omega<0$ and $\omega>0$. Actually, for a given model, $L_0$ and $|\omega|$, we observe that a difference between ingrowing ($\omega<0$) and outgrowing ($\omega>0$) systems arises only at long times ($t \gtrsim t_c=L_0/|\omega|$), when the expanding surfaces cross over to the statistics of curved KPZ systems, whereas the shrinking ones become completely correlated. A generalization of the Family-Vicsek scaling for the roughness of ingrowing interfaces is presented. Our results demonstrate that a transient flat statistics is a general feature of systems starting with large initial sizes, regardless their curvature. This is consistent with their recent observation in ingrowing turbulent liquid crystal interfaces, but it is in contrast with the apparent observation of curved statistics in colloidal deposition at the edge of evaporating drops. A possible explanation for this last result, as a consequence of the very small number of monolayers analyzed in this experiment, is given. This is illustrated in a competitive growth model presenting a few-monolayer transient and an asymptotic behavior consistent, respectively, with the curved and flat statistics.