Perturbing the Shortest Path on a Critical Directed Square Lattice

TL;DR

This paper studies how small perturbations to the shortest path in a critical directed lattice cause non-local effects, revealing power-law distributions in path length changes and enclosed areas with specific exponents.

Contribution

It introduces a novel analysis of the non-local impact of edge perturbations on shortest paths in a critical directed lattice, quantifying the effect with power-law exponents.

Findings

Power-law distribution for shortest path length differences with exponent ~1.36.

Power-law distribution for minimal enclosed areas with exponent ~1.186.

Edge perturbations have non-local, scale-invariant effects on the lattice structure.

Abstract

We investigate the behaviour of the shortest path on a directed two-dimensional square lattice for bond percolation at the critical probability $p_c$ . We observe that flipping an edge lying on the shortest path has a non-local effect in the form of power-law distributions for both the differences in shortest path lengths and for the minimal enclosed areas. Using maximum likelihood estimation and extrapolation we find the exponents $\alpha = 1.36 \pm 0.01$ for the path length differences and $\beta = 1.186 \pm 0.001$ for the enclosed areas.