# Critical phenomena of a hybrid phase transition in cluster merging dynamics

## TL;DR

This paper develops a theoretical framework for hybrid percolation transitions caused by cluster merging, revealing the need for two correlation-length exponents and showing that clusters have fractal boundaries but are not fractals.

## Contribution

It introduces a new theoretical approach to hybrid phase transitions in cluster merging dynamics, including finite-size scaling and the necessity of two correlation-length exponents.

## Key findings

- Two correlation-length exponents are needed for giant and finite clusters.
- Finite-size scaling method for HPT is established.
- Clusters have fractal boundaries but are not fractals.

## Abstract

Recently, a hybrid percolation transitions (HPT) that exhibits both a discontinuous transition and critical behavior at the same transition point has been observed in diverse complex systems. In spite of considerable effort to develop the theory of HPT, it is still incomplete, particularly when the transition is induced by cluster merging dynamics. Here, we aim to develop a theoretical framework of the HPT induced by such dynamics. We find that two correlation-length exponents are necessary for characterizing the giant cluster and finite clusters, respectively. Finite-size scaling method for the HPT is also introduced. The conventional formula of the fractal dimension in terms of the critical exponents is not valid. Neither the giant nor finite clusters are fractals but they have fractal boundaries.