Evolving criticality in iterative bicolored percolation

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 and transform criticality. 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 criticality, in which scale invariance persists across generations.

Figures (3)

• Figure 1: (a) Schematic of the IBP process. Starting from a configuration with a two-state critical structure, where adjacent clusters have distinct states (colors), each cluster is independently recolored, and neighboring clusters of the same color are merged to form the next generation. (b) In the loop representation, the IBP procedure corresponds to the stochastic elimination of loops (cluster boundaries). The loops surrounding the origin $O$ are the so-called nested loops.
• Figure 2: Evolution of fractal dimensions $d_f(m)$ under the IBP process with $p_m=1/2$ for all generations $m$. (a) Results starting from O$(n)$ loop configurations at $x_{-}(n)$. (b) Results from fuzzy Potts configurations. Curves represent theoretical predictions. Insets: deviation between fitted and exact values of fractal dimension, $\Delta (m)=d_f^{\,\rm fit}(m)-d_f^{\,\rm th.}(m)$.
• Figure 3: Scaled size of the largest cluster, $C_1(m)/L^{d_f(0)}$, across generations $m$ for various system sizes $L$, starting from the critical Ising configuration. Lines represent theoretical predictions, $\sim L^{d_f(m)-d_f(0)}$. Data are normalized so that the first point of each generation equals unity. Inset: cluster number density $n(s,L)$ at $m=1$, defined as the number of clusters of size $s$ per site. The solid line indicates the scaling $\sim s^{-\tau(m)}$, with $\tau(m)\approx 2.012$. Using the relation $\tau(m)=1+d/d_f(m)$, this value is consistent with the theoretical fractal dimension $d_f(m)=1.97565\ldots$.