This is an example of a phenomenon called universality . What it means in this case is that the structure of the percolating cluster doesn't depend on the details of the model that you use to study percolation. This is a very deep property of physical systems near phase transitions. It underlies a concept called the renormalization group in particle physics and the epsilon-expansion in condensed matter physics. In 1982 Ken Wilson won the Nobel Prize by exploiting these concepts to invent a way to make calculations with phase transitions.
Exercise Think about the percolation problem in two dimensions with a triangular lattice.
Write a code to simulate it. Exercise Examine universality by calculating different quantities in several different two-dimensional models. Further Reading Introduction to Computational Physics , Giardino. Introduction to Percolation Theory , D. Stauffer. W.H. Press et al.