Evolving fractal dimensions in iterative bicolored percolation

TL;DR

This paper introduces an iterative bicolored percolation process that preserves criticality while evolving fractal dimensions, revealing a new geometric mechanism for scale-invariant structures across generations.

Contribution

It demonstrates how critical percolation configurations can be iteratively coarse-grained to produce a hierarchy of critical states with changing fractal dimensions, using conformal loop ensemble analysis.

Findings

Exact generation-dependent fractal dimensions derived

Monte Carlo simulations confirm theoretical predictions

Different initial states lead to distinct critical exponents

Abstract

Criticality is traditionally regarded as an unstable, fine-tuned fixed point of the renormalization group. We introduce an iterative bicolored percolation process in two dimensions and show that it can both preserve criticality and transform fractal dimensions. Starting from critical configurations, such as the O$(n)$ loop and fuzzy Potts models, successive coarse-graining generates a hierarchy of distinct yet critical generations. Using the conformal loop ensemble, we derive exact, generation-dependent fractal dimensions, which are quantitatively confirmed by large-scale Monte Carlo simulations. The evolutionary trajectory depends not only on the universality class of the initial state but also on whether it possesses a two-state critical structure, leading to different critical exponents starting from site and bond percolation. These results establish a general geometric mechanism for evolving fractal dimensions, in which scale invariance persists across generations.