Dynamics of driven interfaces near isotropic percolation transition

TL;DR

This paper investigates the behavior of interfaces near the percolation threshold in random media, revealing different scaling regimes and connecting interface dynamics to percolation properties through simulations and theoretical analysis.

Contribution

It introduces a scaling framework linking interface roughening near percolation to percolation cluster properties, supported by Monte Carlo simulations and theoretical insights.

Findings

Interfaces are self-affine away from the transition, consistent with KPZ universality.

Near the percolation threshold, early-time behavior differs and can be scaled from percolation properties.

At depinning, the interface loses self-affinity, indicating a transition in behavior.

Abstract

We consider the dynamics and kinetic roughening of interfaces embedded in uniformly random media near percolation treshold. In particular, we study simple discrete ``forest fire'' lattice models through Monte Carlo simulations in two and three spatial dimensions. An interface generated in the models is found to display complex behavior. Away from the percolation transition, the interface is self-affine with asymptotic dynamics consistent with the Kardar-Parisi-Zhang universality class. However, in the vicinity of the percolation transition, there is a different behavior at earlier times. By scaling arguments we show that the global scaling exponents associated with the kinetic roughening of the interface can be obtained from the properties of the underlying percolation cluster. Our numerical results are in good agreement with theory. However, we demonstrate that at the depinning transition, the interface as defined in the models is no longer self-affine. Finally, we compare these results to those obtained from a more realistic reaction-diffusion model of slow combustion.