Iterative 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 black or white sites that are randomly placed, we randomly reassign the color of each percolation cluster and obtain coarse-grained configurations by merging clusters of the same color. It is shown that this process can be infinitely iterated in the thermodynamic limit, leading to an iterative percolation model. Further, 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. Finally, following a similar process, the iterative percolation is defined for critical bond-percolation clusters.