Self-Affinity in the Gradient Percolation Problem

TL;DR

This paper investigates the self-affine scaling properties of the front in 2D gradient percolation, revealing a universal Hurst exponent and a crossover length scale beyond which the front behaves like uncorrelated noise.

Contribution

It demonstrates that the percolation front exhibits self-affinity with a specific Hurst exponent and identifies the crossover length scale, clarifying the nature of multi-affinity and overhang effects.

Findings

Hurst exponent of 2/3 for the front's self-affinity

Crossover length scale proportional to gradient^(-4/7)

Self-affine behavior persists after removing local jumps

Abstract

We study the scaling properties of the solid-on-solid front of the infinite cluster in two-dimensional gradient percolation. We show that such an object is self affine with a Hurst exponent equal to 2/3 up to a cutoff-length proportional to the gradient to the power (-4/7). Beyond this length scale, the front position has the character of uncorrelated noise. Importantly, the self-affine behavior is robust even after removing local jumps of the front. The previously observed multi affinity, is due to the dominance of overhangs at small distances in the structure function. This is a crossover effect.