Perturbing the Shortest Path on a Critical Directed Square Lattice

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.