Invading interfaces and blocking surfaces in high dimensional disordered systems

TL;DR

This paper investigates how invasion fronts in high-dimensional disordered systems change behavior as dimensionality increases, revealing qualitative transitions and the emergence of mean field behavior at high dimensions.

Contribution

It identifies dimensional thresholds where invasion dynamics shift from avalanche-dominated to annealed processes, and provides numerical evidence for critical exponents across dimensions.

Findings

Avalanches become flat above 6D, indicating a transition to annealed behavior.

Critical exponents in 6D closely match Cayley tree predictions.

High dimensions exhibit strong fluctuations affecting avalanche size distribution.

Abstract

We study the high-dimensional properties of an invading front in a disordered medium with random pinning forces. We concentrate on interfaces described by bounded slope models belonging to the quenched KPZ universality class. We find a number of qualitative transitions in the behavior of the invasion process as dimensionality increases. In low dimensions $d<6$ the system is characterized by two different roughness exponents, the roughness of individual avalanches and the overall interface roughness. We use the similarity of the dynamics of an avalanche with the dynamics of invasion percolation to show that above $d=6$ avalanches become flat and the invasion is well described as an annealed process with correlated noise. In fact, for $d\geq5$ the overall roughness is the same as the annealed roughness. In very large dimensions, strong fluctuations begin to dominate the size distribution of avalanches, and this phenomenon is studied on the Cayley tree, which serves as an infinite dimensional limit. We present numerical simulations in which we measured the values of the critical exponents of the depinning transition, both in finite dimensional lattices with $d\leq6$ and on the Cayley tree, which support our qualitative predictions. We find that the critical exponents in $d=6$ are very close to their values on the Cayley tree, and we conjecture on this basis the existence of a further dimension, where mean field behavior is obtained.