Wavelet-based coarse graining for percolation criticality from a single system size.

Scaling analysis is a fundamental tool for estimating critical points and exponents of phase transitions in complex systems, typically relying on numerical simulations at multiple system sizes or scales. However, real-world systems often exist at a single system size, making such analysis challenging. Here, we propose a wavelet-based method to extract scaling behavior from a single system size. Considering two-dimensional random and explosive site percolation, we perform wavelet-based coarse graining and compute high-frequency coefficients across multiple effective system sizes, each of which corresponds to the size of the transformed system at a coarser resolution. In these coarser systems, wavelet energy is defined as the squared coefficients that capture cluster boundaries. We finally demonstrate that average wavelet energies follow a scaling law, enabling accurate estimation of the critical points and exponents, which are consistent with those obtained from traditional susceptibility-based scaling analysis. This suggests that average wavelet energy serves as a susceptibility-like observable in percolation systems. Our findings highlight that wavelet-based analysis provides a new perspective on percolation criticality, allowing the identification of scaling properties from a single system size. Furthermore, this approach is potentially applicable to real-world systems such as brain activity patterns, bacterial colonies, or social networks, where collecting data at multiple sizes is impractical or costly.