Corrections to Scaling for Watersheds, Optimal Path Cracks, and Bridge Lines

TL;DR

This study investigates the scaling corrections of watersheds, optimal path cracks, and bridge lines in 2D and 3D, revealing they share similar fractal dimensions and correction exponents, indicating deep underlying connections.

Contribution

It demonstrates that watersheds, optimal path cracks, and bridge lines have equivalent fractal dimensions and correction exponents, suggesting a unified understanding of these models.

Findings

All three models have the same fractal dimension in 2D and 3D.

They share the same leading correction-to-scaling exponent.

The models are closely related through heuristic and exact arguments.

Abstract

We study the corrections to scaling for the mass of the watershed, the bridge line, and the optimal path crack in two and three dimensions. We disclose that these models have numerically equivalent fractal dimensions and leading correction-to-scaling exponents. We conjecture all three models to possess the same fractal dimension, namely, $d_f=1.2168\pm0.0005$ in 2D and $d_f=2.487\pm0.003$ in 3D, and the same exponent of the leading correction, $\Omega=0.9\pm0.1$ and $\Omega=1.0\pm0.1$, respectively. The close relations between watersheds, optimal path cracks in the strong disorder limit, and bridge lines are further supported by either heuristic or exact arguments.