Stochastic growth equations on growing domains

TL;DR

This paper investigates the behavior of linear stochastic growth equations on substrates that grow over time following a power law, revealing two regimes with distinct correlation and diffusion properties.

Contribution

It characterizes the transition between correlated and uncorrelated interface dynamics on growing domains based on the growth index gamma.

Findings

For small gamma, the interface becomes correlated with diffusion-dominated dynamics.

For large gamma, the interface remains uncorrelated with dilution-driven dynamics.

Long-term and large-scale behaviors differ from non-growing substrate cases.

Abstract

The dynamics of linear stochastic growth equations on growing substrates is studied. The substrate is assumed to grow in time following the power law $t^\gamma$, where the growth index $\gamma$ is an arbitrary positive number. Two different regimes are clearly identified: for small $\gamma$ the interface becomes correlated, and the dynamics is dominated by diffusion; for large $\gamma$ the interface stays uncorrelated, and the dynamics is dominated by dilution. In this second regime, for short time intervals and spatial scales the critical exponents corresponding to the non-growing substrate situation are recovered. For long time differences or large spatial scales the situation is different. Large spatial scales show the uncorrelated character of the growing interface. Long time intervals are studied by means of the auto-correlation and persistence exponents. It becomes apparent that dilution is the mechanism by which correlations are propagated in this second case.