# Motion of discrete interfaces in low-contrast random environments

## TL;DR

This paper investigates the asymptotic motion of discrete interfaces on a lattice influenced by random perturbations, establishing stochastic homogenization results under certain conditions and highlighting limitations with ergodic perturbations.

## Contribution

It introduces a stochastic homogenization framework for interface motion in random environments and demonstrates conditions under which homogenized velocities emerge.

## Key findings

- Homogenization of interface velocity for stationary, α-mixing perturbations.
- Counterexample showing non-homogenization for ergodic perturbations.
- Analysis of the impact of random perimeter perturbations on interface dynamics.

## Abstract

We study the asymptotic behavior of a discrete-in-time minimizing movement scheme for square lattice interfaces when both the lattice spacing and the time step vanish. The motion is assumed to be driven by minimization of a weighted random perimeter functional with an additional deterministic dissipation term. We consider rectangular initial sets and lower order random perturbations of the perimeter functional. In case of stationary, $\alpha$-mixing perturbations we prove a stochastic homogenization result for the interface velocity. We also provide an example which indicates that stationary, ergodic perturbations do not yield a spatially homogenized limit velocity for this minimizing movement scheme.