Critical phenomena of a hybrid phase transition in cluster merging dynamics

Recently, a hybrid percolation transition (HPT) that exhibits both a discontinuous transition and critical behavior at the same transition point has been observed in diverse complex systems. While the HPT induced by avalanche dynamics has been studied extensively, the HPT induced by cluster merging dynamics (HPT-CMD) has received little attention. Here, we aim to develop a theoretical framework for the HPT-CMD. We find that two correlation-length exponents are necessary for characterizing the giant cluster and finite clusters separately. 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. A finite-size scaling method for the HPT-CMD is also introduced.