Iterative Site Percolation on Triangular Lattice

The site percolation on the triangular lattice is one of few exactly solved statistical systems. Starting from critical percolation clusters of this model and randomly reassigning the color of each percolation cluster, we obtain coarse-grained configurations by merging adjacent clusters that share the same color. It is shown that the process can be infinitely iterated in the thermodynamic limit, leading to an iterative site percolation model. We conjecture from self-matching argument that percolation clusters remain fractal for any finite generation, which can even take any real number by a generalized process. Extensive simulations are performed, and, from the generation-dependent fractal dimension, a continuous family of previously unknown universalities is revealed.