Fractal dimensions of bridge bonds in directed percolation models

Abstract

Bond percolation is a mathematical model studying the emergence of a spanning cluster as bonds are occupied in Euclidean space without any preferred direction. Bonds are classified into two types: bridge bonds and non-bridge bonds. Bridge bonds are ones that once occupied, a spanning cluster is created in one direction of the system. When the occupation of bridge bonds is prohibited and only non-bridge bonds are occupied, the system is divided into two parts

the boundary composed of bridge bonds forms a fractal object. It has been revealed that the fractal dimension of that object is related to the continuity of the so-called explosive percolation transition. In the problem of directed percolation, however, where the bonds possess a preferred direction of flow, not much is known about these bridges. We obtain the fractal dimensions of the bridges in various dimensions and compare them to those of ordinary percolation. It will further be shown that these bridges relate to the continuity of the phase transition in the same way those of ordinary percolation do, and outline the implications of these results.