The elastic and directed percolation backbone

TL;DR

This paper establishes that the elastic backbone on a cylindrical system is equivalent to the backbone of two-dimensional directed percolation, providing precise measurements of their fractal dimensions and critical points.

Contribution

It demonstrates the equivalence between the elastic backbone and the directed percolation backbone, with detailed numerical analysis and fractal dimension measurements.

Findings

Both backbones share the same fractal dimension of approximately 1.681 at criticality.

The fractal dimension at the edge and for full clusters is approximately 1.841.

The measured fractal dimensions align with DP exponents, differing from previous crossover estimates.

Abstract

We argue that the elastic backbone (EB) (union of shortest paths) on a cylindrical system, recently studied by Sampaio Filho et al. [Phys. Rev. Lett. 120, 175701 (2018)], is in fact the backbone of two-dimensional directed percolation (DP). We simulate the EB on the same system as considered by these authors, and also study the DP backbone directly using an algorithm that allows backbones to be generated in a completely periodic manner. We find that both the EB in the bulk and the DP backbone have a fractal dimension of $d_{b} = d_{B,\rm DP} = 1.681\,02(15)$ at the identical critical point $p_{c,\rm{DP}} \approx 0.705\,485\,22$. We also measure the fractal dimension at the edge of the EB system and for the full DP clusters, and find $d_e = d_{\rm DP} = 1.840 \, 54 (4)$. We argue that those two fractal dimensions follow from the DP exponents as $d_{B,\rm DP} = 2-2\beta/\nu_\parallel = 1.681 \, 07 2 (12) $ and $d_{\rm DP} = 2-\beta/\nu_\parallel = 1.840\, 536 (6)$. Our fractal dimensions differ from the value 1.750(3) found by Sampaio Filho et al., whose value may represent a crossover effect.